I recently got a “new” bike to replace my stolen one, and my one complaint about is that the lowest gear is still too high for some of the hills I ride on. Now, I counted 26 teeth on the largest gear on the cassette. Can I just get a new cassette with more teeth, adjust the derailleur and be done?
Yes, within limits. From your picture I’m guessing you have a short-cage derailleur, which should have no problem fitting a 28-tooth large gear. You can try larger, but might need a longer derailleur.
You might need a longer chain to fit the bigger gear. It’s good practice to replace your chain at the same time you replace your cassette, anyway.
Another option might be to switch out your chainring(s) for a smaller size.