Your best bet is to look on the Craigslist site. In fact, today there's a few good ones in your price range:
boston apts by owner classifieds - craigslist
I saw a 2BR in Medford for $1250 and a few others in your range.
There is also a 2BR in Salem on this site w/heat and hot water included for $1200 a month, so research this site, it's a good one.
I also saw a 1BR apartment in Andover and they want $995, that's quite good in this day and age and there's a picture of the place on craigslist and it's just gorgeous. It's in a real old yet updated New Englander/Victorian home. It would be ideal if you could get something closer to Boston, but I'd even consider this one in Andover if it were me! You'd still be closer to the City via I-93, which can get quite congested, but you have to know when to travel on it, that's all. I googled/mapquested this particular place to the Burlington Mall which is located at 75 Middlesex Turnpike and it's roughly 20 minute away, of course, that's without traffic! I'd still live in this particular one in a heart beat for that type of $$$ a month, you'd be ahead then. Check it out to see what you think on the Craigslist site, it'll bring you right to it, as well as others. Keep checking this site every day too, you'll see them change quite a bit daily and also you'll see all the good ones on here, believe me! You have to try to scoop one up quick this time of year with the college students going back to school next month.
I know you want to live closer to the City, yet easy access to Burlington. Davis Square or Inman Square apartments would be where a lot of the students will want to live I would think due to Tufts being near this whole area, just naming one of the colleges, there's several others of course near those squares. I know from experience, the rents will be jacked up sky high you know, due to that happening.
Another thought for you is Winthrop. Did you think about living near the coast? It's got a few rentals in your range there. I checked the Highlands site and saw them on there. I can't show you the address here due to their restrictions, but just google Winthrop, MA rentals and/or check the Highlands site by googling it and then when you get there, just click on rentals and you'll see them. The MBTA would be here. Another place you should check is Stoneham may be? That might be a good area to live where it's right along I-93 just a few exits more to go for either Burlington or downtown Boston for that matter and there's "T" buses that run thru here where it's not too far from Melrose/Malden at all.
Best of luck where ever you end up living!