First, do learn how to send. Learning to send is much easier than learning to copy. One mistake I made when I was first starting was to send faster than I could copy, which is surprisingly easy to do with a straight key. If you send faster than you can receive, then you risk the other operator sending at the same rate you are, which can lead to disaster. (I terminated my very first DX QSO by switching off the rig in the middle of the QSO because the other op sent faster and faster, and I couldn't copy at all eventually. What a memory!) If you start with a keyer, then you won't have that problem.
I echo Aleksander's advice to start making QSOs, assuming that you're licensed to do so, and to not be shy about asking the other operator to QRS when necessary.
The best way to learn to copy faster and better is to make QSOs. Not only will your receiving speed increase naturally, but you'll also learn how to deal with noise and bad fists, which aren't as common these days as they were when I started. By the way, I recommend not avoiding QSOs with operators with bad fists, once you learn to copy them; some of my most memorable QSOs (and favorite QSL cards) have been with old-timers whose fists weren't the greatest.
Another way to learn to copy faster is to teach yourself to "throw away the pencil", as my Elmer put it, which means to learn how to copy in your head. I taught myself how by listening to Morse code recordings from the ARRL with no pencil handy, and just trying to copy in my head. What is surprisingly effective is to listen to code that is faster than you can comfortably copy. (I discovered an interesting thing when listening to these recordings in the car: when the sound source comes from several speakers, but the sound is all at a single frequency, then there are nodes, places where the sound is louder, and anti-nodes, places where the sound is weaker, which you can explore by moving your head around.)
A fantastic way to practice copying call signs and numbers is to use Morse Runner, the Morse code contest simulator by Alex Shovkoplyas, VE3NEA. Thanks to Morse Runner I can copy contest QSOs much faster than I can copy ragchew QSOs. (This too can lead to disaster in a contest, so be careful.)