Tel aviv is amazing, lovely beach, great food, great bars. Eye candy everywhere. Jaffa just round the corner. Eilat I'd recommend if you like diving/snorkelling, the red sea reefs are awesome. If not diving, there's not a huge amount in Eilat itself. Although there are some good desert tours, can daytrip to Petra quite easily from there aswell.

Personally I recommend Tel aviv to everyone I know. Really vibrant city with a very diverse and youthful population, it's a huge amount of fun.

In terms of passport, don't worry. They don't stamp. You get an entry and exit card instead.