Well, personally, my favorite parts of the city to hang out around are Rothschild blvd, lillienblum (1 street south) and Echad HaAm (1 street north). Several blocks south on Hertzl and you get to the Florentin area, which is not "picturesque" but its interesting and lots of good places to go out at night. Neve Tzedek is also beautiful, and, heading south (But right before Jaffa) is the Old Train station (HaTachana) which has recently been renovated into quite a beautiful and cool area, near there are the Maine Colony homes which also have been restored and have some interesting and unusual history.

Another interesting part to see (not really touristy) are the old templar houses which have been shifted around to expand Kaplan St. (near the IDF HQ), that will be a beautiful area but they are currently restoring the templar houses.

definitely check out Gan Meir and have a drink at the cafe there at the LGBT center.

Most of my time in TLV is spent eating,hanging out in coffee shops, visiting friends and walking around and eating.