First Timer International Trip 2017

You will need an ETA as Aust does not give out visa on arrival.
Do that now.
Min cost either $20 or free.
www.border.gov.au
You will need one for the both of you.
If the partner is also a US citizen, you can do one ETA each.
If not a US citizen, or from a country that is not eligible for the Aust ETA, then have to do a Tourist/Visitors Visa, about $150.
Our ETA is like your ESTA.

Is it a better idea to keep things like shampoo, hair styling products, toothpaste, and other hygiene products in my checked luggage? To bypass the gel/liquid/aerosol rule in Australian customs?
What snack foods are allowed in my carry-on baggage, that I can continue to have with me until I return to the US (to continue to eat on the flight home).
Yes, all your LAG have to go into hold luggage, then you don't have to worry about it being inspected by TSA on leaving US. If you want, you can have them with you but in a clear bag, and less than 100ml per item. TSA will inspect in the US, Aust on your arrival only worry for drugs, and food, they don't care about LAG.
I have only about an hour and 55 minutes between my Sydney arrival and my departure to Melbourne. I’m concerned about making my last flight and I’m not sure how to get through customs without too much issue. Is there a certain order in which I should arrange my belongings in my carry-on baggage to make it easier for them to be inspected?
Ideally, on arrival to Aust, if I were you, I would have left a bit more time in SYD int to transfer to SYD dom airport. The terminals are far apart, about 1km and not walkable, you have to get a transit bus after clearing incoming Aust immi and customs.
You can use something called Smartgate, but you can also use the human staffed counter.
From the US, its not a problem, 99.9% of the time, you will be waved through after having your passport scanned, and then down to bag reclaim, and then through to Qantas or Virgin or public shuttle bus.
Ideally, I would not bring any food, but if you have to, declare them, you will be given an incoming pax card, but Aust does not have outgoing pax card anymore.
Its all a learning experience, isn't it?
Hand passport and incoming pax card to officer at border counter, passport data line scanned, no stamping of passports anymore either, they are given back to you, but keep them in your hand, you will need it for 2nd stage, then reclaim bags, pass quarantine, they might xray or inspect bags, they might not, then you are out into the open, you are on your own now.
If you flew in with Qantas, go far right to the end of T1 to the Qantas transfer bus, or if with Virgin, go far left to the end of the left hand side, to the Virgin transfer bus.
If yo are late, you might get onto another domestic flight, I am not sure which airline you are flying with into Aust, and from Syd to Mel.
Major airline will put you onto next flight, if you are late, for your fault of your own.
Tiger/Jetstar might say you have to buy another ticket.
I have/will have a pill prescription for antidepressants, how do I go about keeping those during the duration of my trip without them getting confiscated?
Have a doctors letter with you for your antidepressants, you will have to declare the medications on your incoming pax card. Better to be honest.
This is my first time on an international trip alone, and I’m a bit nervous about getting lost. Where can I find digital copies of the airport maps that are current and up to date?
Syd airport used to have digital maps, but I think due to security, they were taken off. www.sydneyairport.com.au would be the one to read up on.
I didn't request specific seating when I purchased my tickets, I was suggested to do this in case I was offered to be moved to an upper class, but I still do have a secured seat on each plane, yes?
Yes, your seat you paid for is 99.9% secured, but its unlikely you get to "upper" class, unless for some reason, it happens to you, but don't bet on it, if you are lucky, yes, but most of the time, with a pax with no status, its unlikely.
As far as entertainment, I'm bringing a novel, a sketchbook, a 3DS, an iPod, and my cell phone. There aren't any issues about having these things with me, right?
Using your US iphone will see roaming charges, which can be hundreds, if you use a lot of data. If phone is unlocked, get to an Optus outlet in Syd or Mel airport, and show id, and get an Aust sim, micro is available here too. Personally, I do crossword puzzles to pass time!
Just remember to bring as much funds as you can afford, some in Aud$ cash, and some in the form of 2 ATM cards.

And do not bring anything even related to work/employment, or something that might even tip off that you "might" go job hunting.
If you are under 31, you would be eligible for the Aust W&HV, its more expensive to apply for than the ETA, but W&HV gives you work rights, ETA does not.

You speak English, Oz speaks English, so you're way in front of most of the world. Personally I don't take anything into a foreign country that I would miss if its confiscated. You're only in country for 2 weeks, how much personal care products do you need?, I buy local. You may not get your brand, but for 2 weeks, what does it matter? I've found the US TSA people unpleasant and officious most of the time, but then I'm Australian and do not have that interface in Oz. I'm surprised that at the transfer time between an International flight to a domestic flight, years ago I think it was a minimum of 2 hours, but I would guess if there are no hiccups you'll make it.

Keeping it briefer for you.
. It is airlines that have restrictions on what can be in carry on luggage, courtesy of governments' policies re thwarting terrorism and even with Aerosols, if they are considered inflammable, I think you will find they are a No No for even checked on luggage.

As caballero suggests buy anything there is any doubt about here in Oz., pricing pretty cheap.

. Snacks of the manufactured version are OK if still in wrapping but need to be declared as food as is anything you eat ( there are some foodstuffs that are prohibited so just have a read of information on www.border.gov.au ).
If you are handed out fruit in flight, eat it on board or dispose of it in bins provided in arrivals hall.

Again, I'd not bother on stocking up with a heap for Australia is very Americanised when it comes to all sorts of edibles and much more.

. Prescription Medications have labels on them in Australia and if it is the same in the US, having them as issued will not be a problem and if you have a doctors prescription sheet as back up all the better.

Re Airports https://www.sydneyairport.com.au/ is typical of what you can find online so just google any location followed by Airport for info and when in an Airport, if you know first where you are heading, you will find good directional signs though because there are so many, sometimes you do need to concentrate.

At Melbourne Airport, if you are heading into the city, the easiest way is via the Red Airport Skybus they are called, leaving regularly from just outside the terminal and you will see a Red Kiosk from where to buy a ticket, Kiosk and Buses very hard to miss. They take you into Southern Cross Railway Station and the ticket fee includes a free shuttle out to wherever you may be staying if within CBD area.

If you have booked right through to Melbourne via an agent, check with them re flight details for some people do that and get held in a transit area at Sydney and then moved on to Melbourne along with your luggage so it may be that you do immigration and customs at Melbourne.
An agent may have also processed an ETA for you.

With anything electronic, though most stuff these days is good for 110 - 240V globally, for charging you will need to get an adaptor for our 240V electrical outlets, they able to be bought here too if you do not already have one.

Don't stress for it is all too easy and many Aussies are easy enough to approach and ask a question if you find you are a bit lost - it's as easy as " excuse me mate ( sir for an older chap ) or lady, would you know where to find etc. " , answers being either " yeah pal ", " Whatta language you speaka " or maybe something akin to " B.....ger off ".
.

I'm planning on bringing just a bottle of shampoo, toothpaste, and hair styling stuff that I already owned, and keeping it in my checked baggage. A friend suggested I keep a travel size bottle of toothpaste in my carryon bag, to make use of in between flights to freshen up.
I'm flying with QANTAS, and I was very concerned about the layover time between Sydney and Melbourne, but I was reassured that it shouldn't take too long.
I know to wear slip in shoes, and store all the things that need to be checked by customs in closer reach inside my carryon bag (I.e. medicine, LAG, passport, boarding pass, etc.) so that I can be faster in getting them checked through.
I don't plan on looking for a job while I'm there, my visit is solely to see my partner, and I've been told I can have my cellular carrier allow international calling and text (which is honestly all I need) during my stay there.
I've not seen maps of Sydney Kingsford Smith, so I was not aware that the distance between the international and domestic terminals were that far apart. About how long would it take me to get there if I were sprinting? Or...fast-walking? :'D
The food I wanted to bring included things like granola bars and packaged crackers, not anything too exquisite, but would keep me energized enough to get from plane to plane.
I'm very nervous about all this, I need as many reassurances I can get. I'm already a big ball of anxious. :'DDD

P.S I already have an ESTA/ETA approved by the AUS gov. so that's one less thing to worry about.

Quantas has between 30 to 40 flights per day between Sydney and Melbourne. I'm sure that if you miss the connecting flight due to no fault of your own, they will just put you on the next available flight which might be just 15 minutes later.

How would I go about asking them to do that if that happened? "I wasn't able to clear customs in time", "my plane landed later than initially thought", things like that? And I wouldn't have to pay for a completely new ticket?

I haven't been to Sydney and I haven't flown Qantas yet, so maybe somebody who has can verify this point. But there should be either a Qantas desk or at least a general "Transfer desk" somewhere along your transfer route from arrival gate to departure gate. That is where you go if you missed your connecting flight due to a delay and they will do the rebooking for you. No, you won't have to pay anything if the delay is the airline's or the airport's fault. If you miss your flight because you spent too much time in the duty free shop that's a different story. :)