most people will let you take a car on a test drive round the block with them siting next to you, but as you won't have any insurance it is against the law so don't do it (for your second car if you have fully comp insurance on a current car you are insured to drive any car so can take it for a test drive... but not your first one!)
yep you will have to send the log book back to the DVLA to register the car in your name, but this doesn't stop you getting insurance on it, but you can't drive it until you have sorted out the insurance.
Find the car you want, take somebody with a bit of knowledge with you, agree to buy it then sort your insurance out and then you can go pick it up and drive it home and send the log book off to get it registered to you.