Yeah, at 63,000 miles I would go ahead and change the trans, brake and coolant, just to be on the safe side. 60k miles is usually the mileage most cars have that type of maintence done. Your owners manual should have the recommended maintenence intervals for all your fluids.
If the transmission fluid looks to be nice and reddish pink, as it should, no need to hurry. Coolant I would definitely change. Brakes, I don't flush my brake fluid as its not in my owners manual, but a lot of people here suggest flushing the brakes.
And yes, you want to change out all the trans fluid with a flush. The drain and fill method doesn't remove that much.
Dealerships will charge anywhere from $149 to $209 for a trans. flush. Chains (i.e Goodyear) around $129 and independents a little less... maybe $99.