Bike fit can be complicated. It needs to take into account the kind of rider you are, your flexibility, and your body dimensions. To determine your proper handlebar height, first you need to determine the correct saddle height, saddle fore/aft adjustment, and stem length. You can spend money on a professional bike fit, but there are some rules of thumb that can get you started.
I am kind of old school, but I think the following heuristics are useful.
When you sit on the saddle, is your leg fully extended at the bottom of the stroke if your heel is on the pedal.
Seat fore/aft adjustment:
Is your saddle positioned such that a plumb-line dropped from your knee (actually, the notch right behind your patella) would be in line with the pedal spindle when the cranks are at the 3/9 o'clock position? Adjust this by sliding the saddle forward/aft on the seat-post.
If you are a power rider it is OK to be a little behind the spindle, if you are a spinner, you can be a little in front.
(NOTE: You can adjust this situationally during a ride by sliding forward/aft on the seat. For example: Slogging up a long climb? Shift back in your seat and get more power. Spun out on a descent? Shift forward and get your spin on.)
When you are in the drops does the handlebar obscure the front hub? If you can see the hub, in front of the bars, your stem is probably short. If you see the hub behind the bars, it is too long.
I am going to make an assumption that you are not a racer, at least a road racer, since you just picked up your first used road bike with proper dropped bars. There is some degree of personal preference here, but for starters, I would suggest setting them at the same height as the seat. You can go up or down from there based on your comfort level and time in the saddle.
Keep in mind, that as you make some adjustments, the others may fall out of spec and you will have to go back and recalibrate. For example, if your saddle height is spot on according to the heel method, but you discover that your saddle is too far forward, when you adjust the saddle fore/aft, you might need to go back and look at saddle height again and if the bars still obscure the front hub from the drops. Same thing with stem length and height, but in my experience, less so.
Limitations based on frame/fork:
There are some relatively inexpensive work arounds if your bike is too small or your steerer tube on your fork has been cut below where it will provide a good fit. Many of these look kind of dorky, but we are looking for functionality, not trying to win any cycling fashion prizes.
Stems can be purchased with angles up to 45deg and lengths up to 120 mm.
Long seat-posts can be purchased (I think I have seen some 450 mm seat posts).
Curved seat-posts can also be purchased to help you get the best fore/aft position. Triathlon seat-posts can be purchased that curve forward to get you over the pedals more. Mountain bike seat posts, particularly one made by Thompson can be used to get your seat further back.
Some other notes: You will have more power/endurance/speed if you use diaphragmatic breathing while riding. This means that your positions, particularly on the tops or the hoods needs to allow you to let your belly sag down for those deep breaths. When you are in the drops, you are trying to avoid fluid drag and some breathing restriction is acceptable.