Tel Aviv + Istanbul

The city of Tel-Aviv keeps about 5 beaches open throughout the winter, i.e. there are lifeguards present most of the day. (Note: NEVER go into the water if there is no lifeguard !!!!!!). But the beaches will be closed on rainy days and the lifeguards won't let anyone go in the water if the waves are too high (up to 20 ft. in bad weather !). The water temperature in the winter is about 60F.

If you intend on renting an apartment, you could base yourself in TLV taking multiple day trips to Jerusalem, Haifa, etc., all accessible by bus/train but you have enough time to also move bases at least once while you’re in Israel.
In TLV, you can join the bike shared rental program for the duration of your stay: http://www.tel-o-fun.co.il/en/
You should go to the Dead Sea area and/or Eilat on the Red Sea for a few days both for the obvious attractions and for the warmer weather.
Winters are mild in TLV compared to Europe and most of the US, but Jan-Feb are the coldest months and you’re more likely to be sitting on the beach with your clothes on, reading a book, than swimming in the sea.
As for neighborhoods in TLV, once you've looked at a few options in your price range, come back and you can get more specific results. I suggest you begin by looking at anything near Rothschild Blvd. or west of Ibn Gvirol St.

Tel Aviv is a small city geographically - so most of the places you stay you will be within reach of the beach, as well as neighborhood bars and eateries. Just try and stick to the West part of the city.

To be in the center, you'll want to stay south of the Yarkon River, and West of Ibn Gvirol Street. The area adjacent to the river is the called The Old North, as opposed to the urban sprawl north of the river. It's a quiet neighborhood built in the late 40s, older in terms of population, but very quiet, very treen, and on the Park. that stretches on both banks of the river.
Further south, you're hitting the center which is busier and lively, but still has a lot of quiet streets and is full of little cafes.
Going southward from the center, you may also consider the neighborhoods of Florentine (young and lively) or Never Tzedek (more artsy and pricey). Further south is Jaffa, which can also be a great option - though it's not near the center of Tel Aviv, it's got plenty of charm, is becoming very gentrified - so there are plenty of bars and restaurants and cafes.

Jerusalem is only 40-50 minutes away, Haifa about an hour, Acco a bit more than that.

You should probably book your flight on Turkish Air to Tel Aviv, and just have a layover on the way back in Istanbul.

That's great info, thanks yarden!

Just as a reference, here are the places I can afford in Tel Aviv, based on AirBNB results -

http://d.pr/i/8KS

Look good?

Also I have one more question - sometimes when I travel for longer periods, I like to take a class in something. It's a nice way to meet locals and interact that doesn't involve sitting on a bar stool. Is there a class in something cultural & native to Tel Aviv that would be fun? A bread-making class or something like that?

Places like #4, 14 and 17 are in Florentin which is a very nice neighborhood, just like you are looking for. Two shortcomings- a bit isolated from the rest of the city and not near the beach.
http://www.telavivguide.net/Attractions/Neighborhoods/Florentine_Neighborhood_20051127217/
http://en.wikipedia.org/wiki/Florentin,_Tel_Aviv
12 and 19 seem to be near Rothschild Blvd. which I've recommended
http://www.telavivguide.net/Attractions/Neighborhoods/Shenkin_Area_20051127222/

Around #1 is Qerem HaTeymanim
http://www.telavivguide.net/Attractions/Neighborhoods/The_Yemenite_Vineyard_20051130225/

You most obvious choices will be those around #1 and that bunch.
The pink dot near Old Jaffa could also be interesting.

As for courses - there are many such options, but they would probably be conducted in Hebrew. You need to check out Time Out in English once your're here to look for options - there is a large Anglo community here - but I'm not sure that's who you will want to hang out with.

I think I will try to shoot for the Floretin neighborhood if I can. Reading about it now... thanks so much for your help!

Note that Florentin is an old neighbourhood which has been only partially gentrified. Some streets are very run down, with car repair garages, etc, alongside the buildings.

I suggest that you post the address of any apt that you are considering and ask for our opinion. In addition, you can use Google "StreetView" to look at the building and the street.

Florentin was an old working class neighborhood set up partially by Greek Jews who brought their Sea Faring experience to build the port in Tel Aviv.

It was a working, lower middle class neighborhood for ages, and in the 80s became "the place" for young, pot-smoking kids after the army.

In the 90s it was regarded as "the next hot spot" for investors who bought up apartments hoping it would shoot up in value. But that didnt really happen (albeit in the past 6 years the whole city has shot up in value). But Florentine remains a good mix of old and new, traiditional and modern, it's not so clean, not so quiet, but has plenty of charm. And has a ton of restaurants and bars and even galleries and boutiques in recent years.

I'd say the down side is the distance from a decent beach (although it's probably only a 15 minute walk)

http://en.wikipedia.org/wiki/Florentin,_Tel_Aviv