Okay, so the link shows that you have a 12-26T rear cassette. By your brief description, I suspect you may not yet have a good gear-shifting strategy, but you may want to change the cassette as well.
First off, you need to figure out what a good cadence is for you to pedal. This is how fast you spin the pedals. Many people start out trying to muscle through a slow RPM, which is hard on the knees and will burn your legs up in just a few miles (though there are a few people who can ride this way for long distances). Typically, a recommended cadence would be 80-90 RPM. If you don't have a cadence meter, you can count your pedal strokes for 15 seconds and multiply by 4 (or some other combination of factors of 60). How fast you FEEL like you're going is irrelevant. You need to pedal with an effort and RPM that you can sustain. Choose a gear combination that allows you to do this and that also keeps the chain as straight as possible. In other words, don't ride the big-big combo and don't ride the small-small combo--those are both intermediate ratios that can be attained with a different combination that keeps the chain much straighter.
For a charity ride, it will probably be a ride-at-your-own-pace, and there will likely be slow riders. You can train a little harder for a shorter distance in the meantime, and back off your effort a bit on the day(s) of. (I'm assuming the 250 miles is over several days--if that's a single-day event, you're probably out of your league--that's not a casual rider distance in one day).
As for climbing hills, there's no substitute for training. If it's a short, steep hill, you can shift down to a low gear (small in front, large in rear), and muscle through a slow cadence if you end up in your lowest gear. Since your largest rear gear is 26T, you could swap the cassette for something like a 12-28 (a little better for climbing) or a 12-32, 12-34 or 12-36. However, those last 3 may require that you also change out your rear derailleur for one with a longer cage (that's the part that hangs down with 2 pulley wheels and keeps the chain tensioned).
Another option would be to replace your crankset with a compact double (50 and 34 teeth), or go to a triple crankset, which would be something like 52-39-30, but that would then necessitate changing your shift lever to a triple-capable one, and possibly replacing the front derailleur as well.
But, again, there's no substitute for miles in the saddle.
