First off, Welcome to Cap Kelley :wavey:
Now lets take a shot at those questions.
What and how are ways to link to other sites?
Good question, link exchanges can be arranged for on a one to one basis via communities and forums such as CAP. We have an area dedicated to link exchanging right here: http://www.casinoaffiliateprograms.com/bb/forumdisplay.php?f=28
You can also exchange a lot of links quickly at services such as Link Partners
http://www.linkpartners.com
Or you can even buy links at several auction sites online.
Why would sites want to link to a new site?
Well, because they want link partners too or because they are also new sites. Even established sites will often exchange links with new sites because they know in a few months YOUR site wont be new anymore and they will have a potential good link partner. What most ranked sites wont do is exchange Front Page links with you, until you have been around for awhile and have established PR (Page Rank) dont bother asking the biggies for front page exchanges.
Is it free?
Generally it is unless you purchase links.
How do you do banner exchanges?
These arent as common as they once were and from an SEO standpoint are probably not as effective as a well written text link. They can be had had thru the same channels I outlined above or you can purchase banner space at sites you feel have the type of traffic you are looking for. Take a look here if you just gotta buy banner spots http://www.adbrite.com
Hope this helps and good luck! :thumbsup: