I have been doing w2w for the last 15 years. I used to be very careful and, frankly, very slow on the first lap, but not anymore. For me, it is all about:
A. Trusting your own abilities - "I can drive with cold tires, or in the rain, or with worn-out tires - hell yeah, here we go!"
B. Trusting the car and the tires - If you have warmed the car properly on the out lap (and you didn't have to sit too long on the grid), you should have enough grip to give 85% or so.
C. Know the car and the tires - On track days and training days, I try to do the same routine as I do in a race, every session. I heat the tires on the out lap and I hit as much pace as I estimate the car can do as soon as I get the green flag. Or a bit higher pace, just to learn where the limit is. Building experience.
D. Know that bumping someone is not the end of the world - Don't get me wrong, I am NOT the driver that wrecks other peoples cars and hasn't been in any class or series. But, with some years of experience, I have also significantly lowered my respect level for car-to-car contact, bumps, trading paints, etc. Usually, you know who has a freshly painted car and who is racing on a shoestring budget and of course, you are extra careful around those guys. But bumping your mates car a little because you gave it a few % too much... not the end of the world.
I used to race in a series of vastly mixed cars, from super7s (which I drove) to M3s, 911, Lotus, Corvettes, tube frame cars, etc. I got the impression that the fellows with massive, wide tires wouldn't get either temp/grip or confidence in their cars until well into the race, lap 3, or for some even lap 4. That gave us with smaller cars and smaller tires a nice advantage. Weirdly, this was extra prominent in the rain! It should reasonably be the opposite since heating up a rain tire in the wet isn't much of a task, but it wasn't. Maybe it had more to do with confidence and finding the limit (this was obviously an amateur series), but the gang of little Westfields, Caterhams, Furys and such would take off like a swarm of bees as soon as the lights went out (drifting a Westfield in the rain is better than sex, by the way. Like... better than really GOOD sex!), while the larger and heavier cars would need several laps to get up to pace.