I agree with many of the options provided by others; however, I think a great fit might be Arlington, Brookline, Newton or Jamaica Plain. These three areas each have unique qualities, but all should offer options with outdoor space, easy access to trains or bus, as well as giving you a bit more for your money. You will be hard-pressed to find a rental in Charlestown with outdoor space, especially if you're looking for private space.
Please let me know if you have any questions!
As far as transit options you will have a combination of car, bus and rail options in most of the above communities. I would recomend you meed with a Realtor that specializes in rentals and explore the difference communities. Rents are a bit of a shocker depending on where you are relocating from. Rents are way up and many renters are converting over to buying a home as interest rates are super low and in their favor. The question of " Why pay a landlord when I can pay myself " seems to make sense in our real estate market.