Lowell is pretty far. You can find a place much closer. The "cheapest/safest" places to live in Boston are Allston and Brighton. PM me if you want to know what areas to aviod (crime, violence, ect.)
I lived in allston/brighton for four years, moved to Somerville and then to west Medford. The neighborhoods, rental rates and space got better with each move. Don't use an agent to find your place it's a waste of money. Use craigslist
Allston: Unless you like being surrounded by undergrads, bars, loud evenings, paying too much to rent a small place, searching for parking forevert and crappy public transportation (green line) don't go here. From my experience and that of my friends Allston is not a place to move if you're concerned about safety (who isn't?). There are definitely areas to avoid in Allston, but nice areas exist. Don't move to Roxbury while we're on the topic. Brighton is better than Allston. but you're commute will suck.
Somerville: Awesome. I recommend Davis Square. Davis has a few bars, nice restaurants Right on the Red Line so it's a short commute to Boston (10 - 15 min). Always felt safe and it has an Anna's Taqueria (you'll find out more about Anna's later). I got a much bigger place that I would have paid a lot more for in Allston and a parking spot.
Medford (West Medford): My place in Somerville was a block from Medford so it's pretty much the same scene as Somerville, but a little further away from boston. I recommend West Medford. It is one commuter line stop from North Station in Boston (green and orange lines), takes eleven minutes to get in the city (by rail) and costs the same as regular T fare. Rent is not Lowell cheap but you'll find your self a nice place.
Overall, if you're new to Boston I'd go with Davis Square and work your way from there.