What cuts of meat are preferable when you want to avoid drying out?
1) Chicken is a tough meat to cook right, in that it dries out very easily. A few too many minutes of cooking and you can go from a deliciously juicy piece of chicken to a dry and chalky piece of chicken.

If you're newer to cooking chicken, I highly recommend using a meat thermometer and taking the meat off the heat about 5-10 degrees before the final temperature is reached, as the meat will continue to rise in internal temperature during the resting period.

For cuts of chicken, it doesn't really matter. I personally like chicken breast, which is white meat and is lower in fat. But chicken thighs (dark meat) are also great, and tend to be juicier by default because of a higher fat content. The entire chicken is great!

Start with some known chicken recipes, which include cooking times and spice/marinade combinations. Once you try a few, you'll get an idea of how to experiment on your own.

Chicken Breast Recipes

2) It will depend how you intend to cook and eat the meat. If you just want a piece of beef, you're going to usually want what's traditionally considered a steak. This will be a NY Strip, Rib Eye, Tenderloin (Filet Mignon), Porterhouse, and many more. Stay away from Chuck (stew beef), Round, Flank, Sirloin etc. These cuts are tougher and require more than just grilling or baking to become enjoyable (although can be very enjoyable when marinated, or slow cooked).

Well done meat isn't always tough - I highly suspect it was the cut of meat you were trying to prepare in your other question. Although, most people will find well done meat less enjoyable than, say, medium-rare or similar. This is preference though - so eat what you find most enjoyable. (although if you have not tried a medium-rare NY Strip or Porterhouse, I recommend it!)

3) This entirely depends on how you cook the meat and what cut of meat you start with. Things like Chuck are usually cheaper than NY Strip, because they are tougher to start with, and contain more connective tissue that must be broken down before it can be enjoyed. For example, Sirloin is usually a bit tougher than a NY Strip, however if marinaded and slow cooked properly, it can be turned into a delicate, tender and delicious Tri-Tip! Same goes for Flank, which when marinated properly can be turned into delicious carne asada for "street tacos".

For more tips on how to prepare and cook your meat, see your other question: What am I doing wrong when cooking meat?

If you have the ability to find it, I highly recommend watching some episodes of Alton Brown's Good Eats tv show. It's fantastic, and doesn't just explain how to cook something - it explains the why reasoning behind cooking things a certain way. This will equip you with the knowledge necessary to experiment on your own without a recipe!