Takes time and effort to develop the strength to support your upper body with the hands and arms without pain. I had the same problem with the road bike I got in June -- my first road bike in more than 30 years. Two years of riding hybrids didn't condition me for drop bars.

I began with the handlebar elevated to saddle height. Then added thick foam padding. Meanwhile I worked on arm and upper body conditioning.

By September I was able to lower the bar and remove the thick padding. Road buzz through the thin bar tape was still uncomfortable so I used lightly padded bar tape.

There's still some occasional back and neck pain but that's from an old car wreck injury. But the elbows, wrists, hands, etc., are okay.

If your upper body and arms are completely untoned, start gradually and easily with lots of stretching and little or no weight when flexing and extending the arms -- curls without weight, overhead extensions, etc. Wall pushups are a good way to start with strengthening triceps, pecs, etc. Gradually move to a little weight -- only a couple of lbs, no more than 5 lbs. Then regular pushups, etc. Ditto, abdominal and back strengthening. Start easily, gradually work up.

If your arms and upper body are reasonably well toned from other exercising, a job with manual labor or gardening, etc., you can progress faster in resistance and weight training.

40 years ago when I was in peak condition I could do 50 consecutive pushups, 100 pullups and 200 situps without a break. Those days are long gone. After a car wreck in 2001 broke six vertebrae in my back and neck I did almost no exercise for more than 12 years, other than walking with a cane. So it took a long time to get back into reasonable condition for a road bike with drop bars. But if I can do that over a summer's conditioning at home without special equipment, you can probably get into shape much quicker.