1.08 is not so bad for someone coming in with no previous racing experience at all. With more experience of the track and the car you should be able to make a steady progression, and bear in mind that some of the guys and gals are insanely fast so don't get discouraged if you see drivers that are 10 seconds faster than you. A consistent 1.03 is pretty competetive if you just stay on the track.
iRacing rewards clean driving by adding points to your Safety Rating (SR), so take your time getting the feel for the car and what you can do with it. When you reach an SR of 3.0 or more, you'll also be able to drive the Advanced Solstice which is basically the same car with a better set up, and once you reach 4.0 you get to try the Formula RT2000 Skip Barber car.
For the ovals the same principle applies. SR = or > 3.0 and you can drive the Advanced Legends car, SR = or > 4.0 and the Late model becomes available.
To earn SR you have to join Qualifying, Time Trial or Race sessions, but you don't necessarily have to do them all, or do them in any particular order. If you want to go straight into a race you can do that, you don't have to qualify or do a time trial. From what you write it sounds like you are doing the right thing regarding looking for and joining sessions.
The number of racers seem to vary depending on day of the week and the time of the day. The day of the week with the least number of drivers seem to be Wednesdays, with the week-ends attracting more drivers. From my experience it seems like most drivers are online in what in Central Europe is the evenings, and I suspect that the US racers are starting to come online as we Europeans have to go to bed!
At the moment I think the membership base is relatively small, accounting for some races not being full. At the same time there are really too many races and too many series for the amount of current members but that is to make sure that members won't have to wait hours for the next race to start. We must also consider that iRacing is very new and was only launched about a couple of months ago.
It might be an idea to look in the iRacing forums, or post there for someone who would like to run practice sessions at the times you are online. I've had a few practice sessions with some of my good iRacing friends and it is generally very good fun and can also be a great learning tool when you start talking about how to shave those seconds off your laptimes. We've run in the practice sessions and if someone from outside our group joined the session we'd generally tell them what we're doing and that they are more than welcome to join in the fun.
Which Club and time zone are you in? we might meet up on a practice server and I could give you a tip or two if the personal schedules and time-zones permit.

On the iRacing.com web-site main page there is a menu bar called "Reference" and under that menu you will find user guide, which is, well, a user guide. You will also find the sporting code there, and that one explains a bit more in detail about how to progress. For example it mentions that there is a minimum participartion criteria of two races or 4 time trials in any given season to advance to higher licenses.
I hope this helps somewhat and if you got more questions, never be afraid to ask!