If you connect the phone to a computer, you can copy across things like photos (the folder may be called Camera, Pictures or DCIM) and music. Anything you downloaded will probably be in the Download folder. Anything on the sd card will be safe as long as you remove it before the reset. If the card is set as default storage for anything (eg the camera), the reset process will treat it as part of memory and wipe it.
There are proprietary backup programs and some devices have them built in. I always favour the file-by-file approach as it gives you more control.
Your Contacts and Calendar should be sync'd with Google, but log in on the computer and check that. When you sign back into Google, they should simply re-appear.
After a reset, run the phone vanilla, with no extra apps or content, for a while and see if it behaves. If it does, add your apps back one by one. If the problem recurs, you've nailed the culprit if it's an app.
On top of storage, the device also has an area of working memory and this is also fairly limited. Overloading it can cause the issues you're having, so you could also try tapping the recent apps button (to the right of the home key). If there's a "clear" option - sometimes an X - tap that and it'll clear a lot of stuff out. The OS should handle memory for itself, but some apps are better and handing resources back than others. My elderly Samsung tablet often needs a restart after a day or so of use.
