You do have to give notice, but to give notice you need to know exactly where you're getting married as your notice is venue specific.
Notice of marriage is £42 per person if both UK nationals or you have settled status. Its £57 if you require home office referal. So the cost really depends on your immigration status.
A straight forward notice means you can give notice and get married 29 days later. With a more complex notice where the home office are involved they have the powers to extend your waiting period to 70 days.
Witnesses really can be anyone, you dont have to know them personally. If you travel somewhere to get married, and stay at a hotel, the hotel staff are normally very obliging.
For what its worth, I had a tiny a wedding, just immediate family and I absolutely loved it. It was perfect for us.
As a registrar, the small ceremonies are my favourite. You find these are the couples that want the marriage not the wedding and they are really lovely, much more meaningful than the big ott ceremonies.