I was the same when I went for my session for fitting - as soon as the lens got anywhere near my eyeball I couldn't help my eyelids violently closing shut.
After something like about 20-30 mins of the optician trying he was starting to get annoyed with me and told me off, if I wouldn't let him fit the lens I could never wear them. After many struggles he eventually got a pair in and asked me to try wearing them for a little bit (I think about 30 mins). My eyes were in agony but I just about managed to survive.
When I had to start doing it myself, I had to get up something like 30-60 mins early as it took me that long trying again and again to get the lenses in and my eyes would be hurting for another 30-60 mins until they would settle down, but I gradually got better at it eventually (can't remember exactly how long it took me as this was I think 13 years ago).
Gradually my eyes have become less and less sensitive, these days I can easily put in my lenses on first attempt without using a mirror, and even if my hands/lenses are a tiny bit dirty my eyes don't sting (although I know it's important to keep them clean).
Tiny tip (I've only tried this using specsavers solution for soft lenses), if I don't succeed on the first attempt or two, I will wash out the lens to get rid of any dirt, and to lubricate the lens. Also when I started wearing lenses, if I had been smoking prior to putting in the lens my eyes would sting for quite a while. Doesn't happen these days, but used to be a major consideration when I started wearing lenses. Also if I get dry eyes, instead of using expensive eye drops, I just use a little contact lens solution - my thought being that it's mainly sterile water with extra stuff to lubricate and sterilise so I doubt there's any issue doing this. When I wore monthly lenses, I found that I normally would get 3 bottles of solution with 3 months lenses, but I would never use all the solution so instead of paying for expensive eye drops, I could just use solution I already had.
Also another tip - I get my lenses from Daysoft. About £13 to get a month worth of daily disposables delivered to my door. I decided to wear dailies after I had a couple of eye infections years ago (caused by sleeping with monthly lenses overnight). Since then, haven't had any eye infections.
Another little thing - when I started with the specsavers monthly lenses, I couldn't tell the difference by looking at the lens if it had turned inside out or not. These days I can tell just by looking at the lens, and I can definitely notice the lens is uncomfortable if the lens is inside out. Unfortunately I can't say whether I have gradually learnt the ability to spot that the lens is inside out, or whether the difference is more visible with different lenses.