Welcome to the forum meb94!
Normally I would suggest you fly directly on to another detaination upon arrival but....you arrive on a Friday so that means flights will be at their most expensive.You could take the overnight bus but again you would need to book this in advance as they too get booked up quickly at weekends.It isn't just foreign tourists who head off but locals too at weekends :) Having said all that it is worth researching flights to both Cappadocia and Antalya just in case you spot a bargain.
Another point is that many attractions and museums in Istanbul are closed Monday and Tuesday,so,you could spend Sat/Sun doing the main attractions then Monday see other things that aren't closed.Look for a flight for the Monday evening or take the o/night bus that night.
Taking evening flights or o/night buses makes use of other wise dead time to keep your daytimes free.And of course the bus means saving on a hotel fee that night haa haa!
Just going through your questions now...
1.Göreme is one of the main villages in Cappadocia.Many say it is touristy,well yes,it is but with good reason;it is centrally located and thus convenient for visiting the whole area and the term touristy is rather relative anyway.I mean the village isn't some tacky wild place just...busy but then you can turn a busy street corner and be in a lane totally alone.
The hot air balloon trips are stunning but so are the prices(100-160€).Many visit the region and don't do this yet don't feel their trip was lacking so, as you are on a budget,do not feel you HAVE to do it.However, if you have plans for just one special splurge on the trip then Cappadocia has to be about the best place on earth to do ballooning!
August will be very hot and dusty there so probably 2 full days will be enough.For the trip think along the lines of Capp.2 days,Ist.3-4 days, coast 3-4 days.
2.In August there will be no such thing as a quiet beach as I'm sure you can understand but for a slightly quieter one head to somewhere like Adrasan or Cırali.These are smaller more lowkey villages with instant access to interesting sections of the Lycian Way.You can get a bus connection direct from Antalya otogar upon arrival there.If you are planning to hike any of L.W then be prepared for VERY early mornings ,i.e. c.06.00 and walk until noon-ish then go and chill out.In August is is far too hot to be out walking along the trail.Indeed some would say dangerous due to heat exhaustion.Food for thought there....
3.As young people you are sure to want some sort of nightlife in Istanbul aren't you?So maybe consider staying around Galata/lower Istiklal Avenue area.This is hip and vibrant and safe for sensible travellers.
The other option is historic Sultanahmet.Nightlife here is low key,mainly centred around cafe bars ,no nightclubs there :)It is also where you will find cheaper accomodation,more so then over in Galata area.If here then look for places around Küçükayasofya neighbourhood or Gulhane/Sirkeci.These are just slightly away from the main part of Sultanahmet(5-10 minutes walk is all),more locals and less...in-your-face.
4.Don't bother renting a car whilst here.Buses are utterly amazing and go everywhere at any time.We have a truly excellent transport system here in Turkey.In addition fuel prices are hideous and that(along with the price of alcohol)would wreck your budget.
It may be worth you buying an Istanbulkart which is a prepaid card for all public transport in Istanbul-metro,tram,ferry,funiculars,metrobus etc.This will depend on how much walking you enjoy and how much you think you would use it-Istanbul is a great city to discover on foot.One card can be used by up to 5 people btw,so one is enough for you.
http://istanbulkart.iett.gov.tr/en/istanbulkart/pages/anonymous-card/427
5.Saving money....don't shop at places like the Grand Bazaar or other obviously touristy places,too expensive.Eat like a local,head off main strips to the side streets where you will find plenty of small lokanta's.These are small cafes usually family run ,they are simple plain but spotlessly clean,serve great food and are dirt cheap.And they are full of locals!Avoid like the plague any glitzy gin palaces full of tourists with a barker ouside dragging people in. Expensive and generally naff food .
Almost any place you stay will include a generous breakfast in the room price-even hostels so take advantage of this to fill up.At lunchtime there are endless clean safe street food options for a lunchtime snack or even just buy some fruit and a loaf of fresh bread and have a picnic in a park or on the beach.
Consider buying a couple of bottles of beer from a local shop and drinking them on your balcony rather than in bars,they always charge extra naturally and alcohol is expensive here anyway,especially spirits and imported alcohol.
Don't be afraid to ask for discounts,that is;haggle!Even when things have prices on them you can politely ask if they can'round down'a price.Driving a hard bargain in a gentle tone is far more effective than waving your arms about shouting" too much too much!" ;0)
So now you have some homework to do,thinking about the order you'll see these places,checking up on flight times and prices etc.,and deciding which area of Istanbul you'd like to stay in.If you need any help with any of these then just shout up.
