For your situation, as far as working on your cars and stuff late at night, I do Not recommend a "typical" neighborhood where you are right next door to your neighbors.
As far as places to avoid: avoid northside (just north of downtown), Westside (just north of Orange park, and west of the neighborhoods surrounding the river), and maybe avoid certain neighborhoods in the heart of arlington (around university).
For your situation I would recommend some of th newer devlopments north of Blount Island where the homes have larger lots and spaces between the houses. You may want to consider south Nassau county (yulee, fernandina area). This may put you a little further from your job than you would like, but there are plenty of safe places to live in a more "rural" setting. The same goes for north St Johns county (rural and safe, plus better schools).
It may be tough finding a home with larger lot size in jacksonville with your price range (but fortunately it is a buyers market right now).
Some others on this forum prob have better advice regarding the locations around blount island and northeast jax neighborhoods/developments.
Also search someother threads on here regarding those areas and BI.
Here is on thread for starters about the BI area:
http://www.city-data.com/forum/jacks...land-area.html