I have a bunch of DVDrips at about 600x256 resolution, but when I play them fullscreen on my monitor (1440x900) the quality drastically reduces.
Will increasing the bit rate/resolution of the video help?
If so, what are some good programs I can use?
Once you have encoded, you really can’t do much to improve the video. You are better off, going back to the source video, capture, and encode to what you want.
You could try VLC player. It has a lot of filters and such to smooth pixelated video. As to re-encoding, if the quality isn’t there to start it won’t magically appear.