Oh my goodness yes. You can get rooms in the $20-30 range easily. Check for any motel on Niagara Falls Blvd AKA 62.
As for buses, go to
www.nfta.com for schedules and route info. You can definitely get a bus.
When staying in the Falls, there are some really rough ghetto areas you want to make sure to stay away from. I am not from the Falls myself, but I've noticed things tend to get ugly looking once you get west of 104/Portage Rd. They get worse as you head toward the Falls.
If you have a passport, you'd probably enjoy the Canadian side WAY more & it's much safer. Also, flights into Buffalo-Niagara airport are way cheaper than those into the little rinkydink airport just outside the Falls.
As for nice places to check out in Buffalo, you want to visit Elmwood Ave. As for visiting ANY of this stuff in the winter, I really wouldn't recommend it. Viewing the falls can be downright brutal because of the winds even to someone used to the weather. And the Elmwood strip in Buffalo does its annual death during the winter. By contrast, in the summertime you can literally walk up & down Elmwood and meet all kinds of interesting, friendly people just by simply being there & enjoying a relaxing day.
Another perk of visiting in the summer is the free concerts hosted downtown every Thursday in Lafayette Square. The schedule is posted in May & we get a lot of big name acts. Concerts are accessible by the train downtown. Check back for it: http://www.buffaloplace.com/aboutus/marketing/summerconcertseries.html (broken link)