Hand pain and numbness are a sign that the fit is wrong.

I know you know that, here are some additional tips:

FIRST: The seat height must be correct! This is tricky, as the placement of your foot on the pedal will affect this. If you are running clips, you will want to be in the clips so as to ascertain the seat height. If you are using cleats, set them up so the ball of the foot is just forward of the pedal axle. If your hips rock when pedaling, you are too high, if too low you will feel it on the front of your knees. The right height will make the bike feel lighter. When adjusting seat height, unless you are a mile off, make no more than 1/8"-1/4" adjustments!

The set back is critical too. Too far back will cause more pressure on your hands! Being closer will cause you to be more upright, which takes pressure off your hands. Too far forward messes with your seat height (as does the seat angle) and you may feel constrained, especially in turns.

The seat angle is critical as well. Avoid angling the seat down at all costs! Start at level, and you may find that the seat needs to be pointed slightly nose-high. This is very seat-dependent; the correct seat is different for everyone! If the seat is angled down in the front, it will jam you into the bars and you will have excessive hand pressure. I'm not going into selection of the correct saddle- but that is really important too!

Obviously as others have pointed out, the height of the bars and how far they are from you plays a role too. Generally if you are having hand pain, the bars are too far away and/or too low, causing you to be less upright, and so to support your body, too much weight on the hands. Quill stems were available in various lengths and heights, you have to search one out that does both jobs right at the same time. Generally though, unless racing, go with a higher stem with a shorter reach. I would ignore advice that instructs to avoid raising the bars above the seat height. The correct height will be a position that does not give you hand pain and that's different for everyone.

One point here- dropped bars are for racing. They found their way on to regular bikes though and people think they should be fine for regular riding. They aren't. They are not the most comfortable thing around, simply because comfort has been sacrificed for speed. But if you are on the tops you should be able to set them up so you don't get pain. But if not racing, consider upright bars and a more upright position. This may mean you need a wider saddle if you go this route.

Handlebar padding makes no difference to hand comfort- correct fit does. Padding is only helpful if the fit is right! If you have the right fit, you will find that padded gloves are nice but not mandatory.

A little more on the racing bit- comfort on a bike is all about what you are willing to put up with. Younger people and racers put up with more than older people and people that might want to ride a really long ways. I just got back from something called the Tour Divide, which is a 2700 mile long mountain bike route. I didn't have any hand issues, knee issues, shin splints, back problems, no Shermer's neck- nothing. But I spent a lot of time making sure that the fit on my bike was in fact really correct. That is the key and if you get that right you won't have pain. My point here is that a lot of my comments above might be controversial, but only to those that like to race (even if only to pass someone ahead of them on the bike path) or those who don't realize how much discomfort they are dealing with or have decided that such is OK.