The Weizmann Institute of Science resides in a very nice campus, which is located about 25Km (15 miles) of Tel-Aviv, 55Km of Jerusalem, and 15Km of Israel's main international airport (Ben-Gurion).
A taxi from Ben-Gurion airport to Weizmann Institute should cost 125-140NIS (approximately 40USD), and I advise it. Other options seem a real pain, but details about them and about other things is available HERE.
The Lopatie center has a cafe/restrurant and there are a few add'l eateries on campus, but I prefer three places that are outside the campus at a distance of 50-100 meters from the main gate. Just cross Herzel street and Derek Yavne and you will see Vinona (a cafe at the corner), Ha-Nasich (a middle-easter place next to it). A few additonal house to the same direction (south), there is another cafe.
Tel-Aviv (TLV) prides itself to be a small New York (NYC). I think the claim is justified in the sense that you will like TLV if and only if you like NYC. I happen to love both.
The simplest way to get to TLV is to take a taxi. The cost shoould be 140-150NIS (depending on timing), and if you share it among 3-4 rides this seems comparable to other alternatives in cost, but far faster... I once used a driver called Avshalom (phone: 0523 658441).
A few of the popular outing areas in TLV include
To the web-page of the Weizmann Institute of Science, the web-page of TOC at WIS page, and to Oded Goldreich's homepage.