I recently received a used road bike. I took it to a bike shop where they told me the seat and handlebars had to be raised for my height, so I had them make the adjustments. I now have trouble getting on and off my bike, and when I have to stop my feet barely touch the ground. Would these be indicators that my bike's the wrong size for my height?
While in the saddle, your feet should not touch the ground and your leg should be almost completely extended at the bottom of the pedal stroke. You ought to be able to straddle the top tube fairly easily with your feet flat on the ground.
Sheldon Brown's article on starting on a bicycle may help with getting accustomed to having the seat higher than you are used to. Do visit his site to see the video, but here are the basics:
No you shouldn't be able to fully touch the ground from your seat. Maybe with your toes totally pointed you can get them both down but that's about it.
As for getting on an off the bike, practice will make it easier. If you've never been riding a properly fit bike then I would anticipate it feels odd.
You should be able to place both feet flat on the ground while standing over your top tube with a bit of room to spare so you don't accidentally hurt yourself by landing on the top tube.
As for handlebars they should be at least 1 inch lower than your seat generally. The more flexible you are the lower you can go with comfort.
Your seat should be level with the ground (not the top tube) or slightly pointed down.
Your seat should be high enough that at the bottom of your pedal stroke your knee is slightly bent. If you're experiencing knee pain that doesn't go away after a few rides (to get used to the new setup) the general rule is to move your seat up if the pain is in the front of your knee and down if it's in the back of your knee.
Stem length changes based on your torso length so I have to assume that the shop provided you with the proper length/rise of stem to suit your body.
A proper bike fit is one of the best things to spend some money on with a bike. Just give it a bit of time as you get used to a bike setup properly.
From my understanding (certainly no fit expert), you're probably fine as long as you're not crunching your privates on the top tube. Your leg should be close to fully extended at the bottom of the pedal stroke, so you would be on tippy-toes or less when at a complete stop. You'll also probably find that mounting requires the use of the pedal as a "step" to get on.
Get measured 'properly' and then compare that to your frame - and also consider what you're planning to do on the machine.
If you have drop handlebars (e.g. like a road bike) and might be using them a lot, then you'll expect a different setup in comparison to using straight (or courier bars) or aerobars.
Also different bikes have different geometries, i.e. the lengths of the various tubes. I have a relatively long back and short legs and I know that certain brands are wrong for me (i.e. the frame is too short and I am very cramped over the handlebars because I've a downtube correct for my legs, but a top tube too short for my back).
@curtismchale gives a good set of things you can do yourself, but there really is no substitute for expert advice and measurement here. A few centimetres here and then, a spacer on the stem or longer cranks or dropping the saddle ...