Obviously, the more powerful your computer, the faster After Effects will run. RAM is very important and so is a strong CPU. Due to After Effects writing many cache files to your hard drive and obviously loading video files off it, I'd recommend getting an SSD drive if you can afford it. And since more and more effects run off your GPU, a good graphics card will help you too.My computer cost around $800, box only. I have an Intel i7 2700K, 16GB of RAM, an Nvidia Geforce GTX 560 and a Vertex 3 SSD. Even so, previewing and rendering in After Effects is not instantaneously. It sometimes looks that way because I edit my videos - after all I don't want you sitting there watching the progress bar move across the screen for minutes.
Try to get the best machine you can, stay within your budget and just remember it's normal for After Effects to lag a little bit, depending on the complexity of your effect. Especially adding motion blur and depth of field can really drag down rendering of your composition.
It will depend on the money you want to spend, what you want to film and your experience. If you have never filmed any films or VFX before I recommend just using your phone or borrowing a camera - any camera - from family or friends. Just try it out first and see how you like it, experiment with different editing tools and styles and try out some visual effects. The last thing you want to do is spend $2000 just to find out a week later that you don't enjoy film making after all.DSLRs are good if you know what you are doing. You can get great quality when using a DSLR, but you have to be comfortable with controlling shutter speed, aperture, ISO and all the other settings that will affect your video. You should know about Shutter Angle as well as Pixel Aspect Ratio and Video Modes .
DSLRs also usually cost a bit of money so it's more something to upgrade to once you're more comfortable and want to move on to the next level.
There are lots of good compact cameras available for around $100-$200 that can take good quality 1080p video that are great for beginning film makers and I'd recommend starting there once you know that film making is something you'd like to pursue.
In the vlog I will answer 2 questions from my viewers: