Hi and welcome to Madison!
It seems as though a lot of folks come here looking for east vs. west insights, and there have been a number of good conversations in other threads about the relative merits of each. Since there are rather a lot to look through one by one, I went back through the list and found a few that I recall as being particularly informative or useful:
http://www.city-data.com/forum/madis...west-side.html
http://www.city-data.com/forum/madis...t-isthmus.html
http://www.city-data.com/forum/madis...re-should.html
http://www.city-data.com/forum/madis...ghborhood.html
http://www.city-data.com/forum/madis...nt-advice.html
In all of those discussions, you'll find passionate devotees of the east side and equally passionate supporters of the west side. I won't tell you that one is better or worse than the other, because I don't think that is the case, but it does pay to take the time to find the spot that feels right.
As to neighborhoods to avoid, well, as you'll see in several of the threads I linked above, Madison is not a tremendously cheap place to live. If you find a two-bedroom apartment renting for $600 a month, that's probably a clue that the neighborhood is likely to be an iffy one. Specific areas to avoid are the Allied Drive area, the Badger Road area, South Park Street near the Beltline, one or two small patches along East Washington, and some parts of the north side.
Browse the links above, check out the "variety of useful links" sticky, and see if those answer some of your questions. And if you come up with new ones, c'mon back and ask away!