You're absolutely correct; this is a very challenging technique to pick up, it is 100% normal to have difficulty building speed, and you will get faster as you train your hands.
This is one of those (few) times that I consider Hanon's virtuoso pianist exercises to be actually useful. You want to look at exercise 46 (page 76-77 of this edition). The exercise is more comprehensive than you ask for; it covers all forms of trills, in both hands. With these exercises, you want to start slow so that you can play them consistently and to never push the tempo to the point where you sound messy.
As for technique, you want to make sure that you never ever tense your forearms. Ever. Seriously. Don't do it. If (when) they do tense up, give them a few minutes to relax before continuing. When trilling, there are two major motions. One is fairly obvious: waggling your fingers back and forth. To amplify the finger movement, you use the other motion: rotating your wrist back and forth. It has the added benefit of making it easy to avoid tensing up. And in case I haven't made the point clearly enough, don't tense your forearms. :P
The key to trills is practice. Nobody here can tell you how quickly you'll pick it up, mainly because we have no idea what your goal speed, skill level, practice schedule, or natural aptitude are; however, with dedicated daily practice, it shouldn't take more than a week or two to get your trills to a medium tempo.