I hope you'll be ditching your car.

Given your ages, I would look at Lower Allston or Brighton Center. Very safe, walkable to a lot of stuff. There's trolleys and buses that will get you Financial District in about 40 minutes. There's good eats all around, plenty of fun & games. You really can't go wrong if this is your first neighborhood to in Boston. However - these are mostly student neighborhoods - so at some point, you do 'age out'. Avoid Alpha Management.

East Boston is fairly inexpensive still. The Blue Line will take you straight to Financial District in 10-15 minutes.

The neighborhood doesn't have a lot of fun and games, but you can access it all fairly quickly with the Blue (which is the most reliable line, usually). There is great Latin food, parks and public art. It's fairly safe and very walkable. The biggest negative is it's near the airport, so you'll get some noise pollution.
The other negative is that it's somewhat of an island from the rest of the city, so you can't just 'walk home' if the T isn't running, your only option is Uber. Jeffries Point has more people in your age bracket. Eagle Hill is a hodgepodge. Orient Heights is mostly families. If you know a little Spanish, you unlock a ton of DLC for East Boston.

I would look as far as Revere off the Blue Line. Though it gets less "city" the further you are.

I'd then look at anything off the Orange Line in Malden or Medford. Super safe, both have little walkable town centers with good eats scattered throughout. They're not exactly "city vibes" though.

Also Quincy. Though, it's a little farther away and off the Red Line, which has been really bad lately. Like Medford & Malden, it's super safe and has a walkable town center.

Your price range is fairly low, but for a small 1BR - I'm sure you'll find something. If you push to $2200, you'll do better.