Have a look at this post by Snaike. That'll show you how to start a regular stroke play round.
In order to improve tiers you just have to complete ranked rounds. As you complete rounds your average score will start coming down and you'll begin to progress through the tiers as you hit certain numbers. As a reward you'll earn a small credit bonus for each promotion that you can use to upgrade your equipment or put towards a purchase of your own. Exact numbers for each tier are not disclosed to discourage people from purposely trying to avoid promotions which is a practice known as sandbagging.