If you have the oppertunity find an empty car park or playground and spend some time getting used to your gears.
Start, as Arch sagely suggests, with both gear levers set at 2.
NB: Always keep the pedals turning when changing gears, if you change gear without the wheel turning you might 'crash' the gears when you set off.
This means the chain moves across the gears, making a horrible noise and causing the pedals to slip, rather than powering the wheel when you need it to.
So with your gears set 2L & 2R set off and pick up speed, concentrating only on the 1-6 gears.
As soon as your legs feel they are going round too fast and you are not gaining any speed, change to 3R and you will find your legs spin a little slower but you will start to pick up speed.
Repeat this going up to 4R, 5R & 6R.
As soon as you feel you are going too fast or you are having to pedal harder than you'd like, change back down to the previous gear one at a time until you feel comfortable again.
It's good to practice changing up and down the gears so that you are always in a low gear when you stop. This makes starting off again easier and avoids any gear crashing.
What you are aiming for is for your legs to be spinning at roughly the same rate in every gear and you still having enough energy to push harder to maintain or increase your speed.
The above applies to the left hand gears you'll find using 1L will make you legs spin like roadrunners legs, you'll not have much speed but you'll hurt less climbing hills and using 3L will feel like strong strides, great on flat roads for faster sprints.
I concur with getting narrower smooth or slick tyres, knobbly tyres are too slow on tarmac.
Have fun, and really, don't ever worry about getting off on hills, cycling is not all about punishing ones self.