Rehovot is a city in the south-central part of Israel. While not being very unique or interesting for tourists it is a great transit station because of its location (close to Tel Aviv, the airport and Jerusalem) and because of the high number of students in it.
North towards Tel Aviv
Walk north on Herzel street (the main street) in the direction of the train station. When you get to the train station keep on walking down the road until you get to a big junction. Cross it (the name of the street changes to Weizman street or road 412) and stand at the bus stop. Better use a sign as cars can go to multiple destinations from here. You can also take every form of local public transportation that gets to the train station and walk to the spot mentioned above but it's usually not necessary as it's only 25 minutes walk from the city centre.
South towards Be'er Sheva, East towards Jerusalem, Northeast towards Ben Guryon Airport
Take any public transportation that goes South to Bilu Junction (bus lines 26, 301, 367 and many more/ service taxi). The junction connects the main street of the city (Herzel street) with road number 40 that goes to Be'er Sheva. Stand at the bus stop there and make sure you are at the right place- road 40 southbound- either by asking or by checking the plaque at the bus stop. The chances of getting a direct ride all the way to BS are slim as there a LOT of places on that road but little by little it should be possible. Also possible to get a ride to Jerusalem from the same spot, just ask the driver about where is the next good hitchhiking position. To get to the airport: You need to stand on the northeastern corner of the same Bilu Junction. Once again a direct ride is not very likely but people going to Petch Tikva, Modiin and maybe even Jerusalem will be able to help you. Try asking to get to 'El-Al junction' which is a major junction on road number 40 and is very close to the airport.
For those going on road number 40 and wish to continue north in the direction of Tel Aviv it is better to get directly to "park hamada" ("park of science"). You can do it pretty easily by going to the junction of road number 40 with the eastern detour (Menachem Begin road) that does not enter the city and goes directly to the hitchhiking spot towards Tel Aviv.
Accommodation and Sleep
The entire east side of the city, beyond the eastern detour road (Menachem Begin road) is pretty much deserted and you should have no problem sleeping there.
Other useful info
Toilets: like in the rest of Israel toilets in public places are usually free. In Rehovot there are two shopping malls- a big one (which also serves as the main bus station) and a smaller one a little bit to the north on Herzel street, both with toilets. The toilets in the train station are also free but you need to have a ticket in order to get to them.
The faculty of agriculture: an academic institute in the north exit of the city, near the train station, usually referred to as "hafaculta" (Hebrew for "the faculty"). Inside the premises of the faculty you can find free toilets, 24h free computer room (in the main entrance to the right, you will need a password to log in but you can ask one of the students for his. The wifi is open for everybody) and most important- a lot of students. As a great part of the students in this institute (and in many other institutes) were hardcore hitchhikers/ travellers/ couchsurfers in the past there's a good chance the just by addressing one of them and asking for a place to stay you will get a couch for the night! Good chances of finding Russian speaking students but other languages also possible (mostly French and Arabic). You can also find there free showers and kitchen in the dorms- ask anyone where they are (just notice that there are girl and boys dorms). The kitchen and showers are for an entire floor on in every room so they should be accessible. Once in the dorms you can just use the facilities or ask for permission from one of the students but they would probably don't mind as long as you keep the place clean). The key is NOT to be shy. Bear in mind that, like when entering any other public place in the country, you will probably be subject to security check at the main gate. Just answer their questions and it shoud be fine.
Free food: Near the central bus stations you can find Rehovot's market. If you get there at the end of the day (around sunset on weekdays or late afternoon on Friday) there's a good chance of finding good fruits and vegetables on the ground or in big stockpiles. It is common to search those piles and no one would mind. The market is closed, like almost any othet thing, between Friday evening and Sunday morning.