Any ranked strokeplay round will count towards your average which is what your tier is based on. You need to get your average down to a certain number to advance to the next tier and have played a certain number of ranked games in your current tier. I can't remember what the numbers are for amateur to pro but I'm sure someone will be able to help you.
These rounds can be played on your own or with other players as long as you select ranked round. Alt shot, matchplay, CTTH and blitz don't have any effect on your average. Playing in WGT or country club strokeplay tournaments does of course.
As you progress through the levels and tiers you'll generally find the better clubs have faster meters. They also tend to have better precision, sometimes less forgiveness and more spin along with high trajectory. The meter speed guage gives an idea how fast the meter will move but there are some clubs that have show the same meter speed but play slightly differently in reality. Think of the pointer starting at 9 o'clock on the meter speed indicator, that would be the slowest, 12 would be pretty fast and 3 o'clock would be lightning. Many people recommending learning to play with a fast meter early on as you'll need fast meter clubs eventually but that might make life pretty unpleasant for a while. Personally I loved the Raptures, I think I bought the next higher set whenever I could.
As already mentioned you can slow the meter down by buying balls that have better feel. This is one of the stats listed for each ball, the more dots of feel the more it slows the meter. These balls are more expensive though so playing clubs with a slow meter can save you a few $$ as you can get away with cheaper balls. The Tour SD balls are great value for money at 65 credits for 3, a hell of a lot less than the 600 you have to pay for 3 Max pro level balls.