Snow Leopard is not an update - it's a system upgrade. The DVD is no longer available in the online Apple store or the High Street stores.
You can get it only by phone now from Apple.
In the US, call 1-800-MY-APPLE and ask for a sales assistant. Last quoted price was $19.99 for the single-user and $29.99 for the family licence.
For other countries, check here; http://support.apple.com/kb/HE57
Don't forget to ask for Sales; the tech support guys can't help.
Mac requirements for SL;
Mac computer with an Intel Processor (SL will not run on G-series processors)
Minimum 1GB of RAM (2 or more advisable)
5GB free HD space.
DVD drive for installation.
When you've installed SL it will be 10.6.3. You need to update that to 10.6.8 with the combo installer from http://support.apple.com/kb/DL1399
However, if your Mac is running slow and won't accept application updates, you need to address that problem first. Upgrading won't fix it (unless you completely erase Macintosh HD prior to installation).
It sounds like your HD may be full - how much free space do you have?
Look at the bottom of an open Finder Window where it says "xx items, xxGB available". If that available figure is under 10GB you may start to experience slowdown; if it's under 5GB you're getting perilously close to not booting at all.
Freeing space on your Mac OS X startup disk: http://www.thexlab.com/faqs/freeingspace.html
How to free up my disk space: http://www.macmaps.com/diskfull.html