Sounds like you are new to AE. If so please start here to learn the basics: Basic Workflow
To get the best playback results make sure that the Magnification Factor of your Composition Panel is set to 25% or 50% and the Resolution is set to Auto. Make sure you read up on how to do a ram preview. The technique is slightly different for different versions of AE.
If you have AE experience then we need to know details about your composition, the source footage, the comp size and frame rate, the other assets you are using in the project and details of your workflow. A screenshot showing the modified properties of the layers giving you problems would also help. You can just drag a full sized screen shot into the reply field on this forum to include it in your post. If the suggestions that I have given you don't help then please include details and system info to include the build number of After Effects that you are using down to the last decimal point and the details of your system and your OS.
Every time you make a change to your composition it has to re-render for you to preview a la "RAM Preview". Rick Gerard gave good advice, and I agree if that doesn't work you should post your system specifications.
I am new to AE myself so I've been through the ringer, at some point you might want to consider an upgrade and set-up your hard disk drives. Don't feel bad if you have to upgrade to get to a workable set-up for you. I thought my gaming rig was powerful and had no idea just how much power AE requires. This link goes into great detail about disk set-up for Premiere, but it also applies to AE.
1. Attached is a picture showing where you can quickly change the resolution from Auto/Full and Quarter as needed. This will help speed up RAM Previews.
2. Make sure your work area is trimmed and isolated on the pieces of footage you are working on, this will help speed up previews instead of rendering the entire footage preview for areas you might not be working on.
Hope this helps! Anyone else feel free to correct me as well since I am a beginner too.