Kutaisi is the second-largest city in Georgia
West towards Samtredia
It's pretty hard to walk out of the city in this direction. You can minibus #8 until the large market at the western end of the city and hitch from there.
East towards Tbilisi
Walking from the main bridge, you have to continue for 15-20 minutes, the road will turn left, you will see also a small petrol station. All cars and trucks have to slow down because of turning so they will have the time to see you and stop. You can hitch also inside the city (the main road cross the city) but is worst cause all drivers are already nervous for traffic, and so on.
When coming from Poti just at the beginning of the city you can turn right for the by-pass road. It is obligatory for trucks (mostly turkish which are unlikely to stop), but all other traffic is passing through the city. Most likely it is the same for the other direction. Get out before the city/bypass or try to get to the other end, but don't try hitching along the bypass.
Sleep and accomodation
Around Bagrati cathedral there is a large park you can camp inside. if not, even safer solution is going to gelati monastery (10 km away), far from the city, very calm and quiet, good water and neighbourhood. You can arrive hitching if not, there is a minibus starting behind the main square (theatre) at 19:00 (last one) costing only 40 tetri (20 eurocent) the way back with no reason costs the double but anyway is less than half euro..