Like Factor30, I'm confused!
Your internet browser has
never 'been Google' because Google isn't a browser; it's a search engine.
Your browser is probably Internet Explorer (because that's the rubbish that Microsoft tries to give you by default) or it might be Firefox or Chrome (because they're the most popular alternatives). Within that browser you can set your choice of homepage (which you seem to want to be Google's search engine).
You state that you keep changing your homepage back to Google, so I assume that you're doing the following. (If not, try it!):
Go to Google:
http://www.google.co.uk/
Then go to Tools > Internet Options > General.
Select the 'Use current page' option for your home page.
Click 'Apply' (Perhaps that the bit you've not been doing?)
(The instructions above are for Internet Explorer. Other browsers are usually similar).
If you've correctly been resetting your home page, but it keeps changing, that suggest that your browser has been hijacked by malware. Download, install and run the free version Malwarebyte's Anti-Malware to identify and fix the problem:
http://www.malwarebytes.org/products/malwarebytes_free/
(NB: When installing that program you'll be offered the chance to accept a free trial of the 'PRO' version. Click to remove the tick alongside that option before proceeding).
Also run a full scan with your antivirus software.
Moving on to your other problem:
Websites remember you by placing cookies onto your PC, so that you're recognised on your next visit. Since that's not happening, one of the following must be true:
1. Your internet security settings are too high. (In Internet Explorer, go to Tools > Internet Options > Security to change them); or
2. Websites are advising you that they use cookies but you're clicking on the option to reject them. (Don't!) ; or
3. Your browser, or another security program, is deleting your cookies at the end of each session. (Look for whatever it is that might be doing this).
Chris