Malwarebyes is good as well. Hijack this is good for pointing things out.
You don't have to click anything shiny or out of the norm. For a while it was common to get a Flash ad approved and then swap it with a malicious ad that looked the same after approval. The ad would then deposit malicious software on your machine without you knowing even if you were on a legit site. Most of the time this is transparent to the end user so the author or user of said package can get the info off of your machine he likes before being found out.
To answer directly:
How easy is it to get stuff off someones computer remotely? Depends on how security conscious you are, medium to really easy. Macs are tougher as well.
How easy is it to put stuff on someones computer? See above, once they have control and as long as you are on the internet there isn't much they couldn't do, including listening and watching on your built in web cam.
What are the signs of keylogging? What he said, there normally isn't any signs, unless you consider that a sign.
How easy to read someones emails? In the wild or on your computer? If it is on your computer, see above. In the wild it depends on your providers security. I don't generally leave email on the server when using a POP3 account.
If I take my pc to an expert, would they be able to tell what has been passed and where to? Depending on the type of payload dropped of like as it part of a botnet or a keylogger? A good security expert might be able to tell where but a good payload won't keep track of what was sent. A good internet firewall might have logs of it though but most consumer products don't log much.