As there is no ringroad, all cars passing Łomża have to drive through the city center or at least along one of its main arteries. If it is your final destination, just drive into town, the average walking time to any place is maximum thirty minutes or you can take a bus.
Reach Zambrowska route. Just after a two-lane road ends there is a huge transmitting pole. On the other side of the road you can find a perfect place for hitchhiking south, average waiting time does not exceed 15 minutes. Be aware it is the favourite place for local hitchhikers trying to reach Zambrów, you might find some competition there.
Southwest towards Warsaw
The best option is to reach Konarzyce, a small village en route to Warsaw (take the bus no. 13). It is most recommended to hitch from a bus stop, as cars all along Konarzyce have to drive by slowly. Alternatively, take the bus to Śniadowo (all buses to Warsaw stop in Śniadowo, sometimes you need to remind the driver to stop there). It is about 15 kilometers from Łomża; best to hitch right after the railroad crossing, since cars have to stop immediately before it.
Places to avoid
The place gets somewhat dead at night. Normal precautions (e.g. avoiding dark alleys or groups of young, intoxicated men) should be taken.
There is also a public bus system (about 1.10/2.20 PLN depending on whether you're a student)
Accommodation and Sleep
On the other side of Zambrowska route (just pass the transmission mast: see South towards Zambrów) there is a whole lot of meadows spanning across several kilometers south and east. Just find a place remote from main paths and you should not be bothered. Also, stay away from places that seem to be occupied every now and then (for example broken glass/other kinds of litter, a fireplace and provisional benches).