Welcome Moddyziz...What part of the area are you looking to move to? There's the city of Miami, and also Metro Miami-Dade county (the suburbs)

In city of Miami, the Brickell area is nice...Lots of very expensive luxury condos, but you CAN find some decent apartments that won't cost an arm and a leg...additionally, you can find some of the same near Coral Gables (a small city with a lot of luxury), and even in Coral Gables proper, although don't expect of whole lot of living space in these apartments...You may want to also roll the dice in Little Havana (some parts are better than others), although let me warn you: a Spanish-speaker would fare much better here than a non Spanish-speaker; same goes for Hialeah, a large blue-collar but fairly safe area in Northwest Miami-Dade county

As far as the suburbs, Kendall and South Miami in the Southwest, Pinecrest and Palmetto Bay in the lower SW, Doral in the central NW, Miami Lakes in the upper NW, and Aventura in the upper NE are among safest areas to live. Googlemap all these locations for a better idea of their relation to Downtown, Universities, Job Site, etc.

A parting piece of advice: traffic is a part of life in South Florida, and if you can get a place close to your major destinations, awesome! Most likely, though, you're going to need to find some good radio programs in the morning/afternoon rush hour. Hope this helps!