What size cap should I use? How do I figured that out?
Basically it depends on three things:
1. Your mains frequency (50Hz or 60Hz), and whether you're using half-wave or full-wave rectification (I think you're using full wave). So assuming 50Hz you'll have 100 'peaks' per second, or 0.01 seconds between peaks.
2. How much current you're drawing. 5V at 1A is 5 watts. Allowing for some inefficiencies in the power conversion, that's going to represent around 0.25A at 24V d.c. (we don't need to be very accurate here)
3. How much "ripple" you can tolerate on the rectified and smoothed d.c. You're using a proper regulator later, so 5 volts (or more) shouldn't be a problem.
You then have a formula:
Capacitance(farads) x ripple(volts) = current(amps) x time between peaks(seconds)
Pluggin in the numbers we have, you get a capacitance of 500uF. So 470uF should do, or 1000uF if you really want to play safe.
I'm assuming I can hook my power to the diode bridge and see what the output voltage is to make sure it's lower than the max 35V that this part can handle. Is that right?
Theoretically the maximum voltage you'll get is 25.5V x the square root of 2, or about 36V, minus any voltage dropped across the diodes. But by the time you've allowed for tolerances in everything - your incoming mains voltage could be up to 5 or 10% above nominal, the brick voltage might vary with load, and not be totally accurate in the first place, and so on - you should be looking at rating everything for at least 45 volts.