This just popped up in Active convos - I had to use clexane for the first couple of trimesters, so not exactly the same scenario. I am the world's biggest coward when it comes to needles though, so I tried everything to make it easier and thought I'd share what I found:
I found it easier if I or DH did the injection, rather than the midwife. We went for slow and steady, rather than the fast approach.
If you rub (or in any way fiddle with) the injection site after the injection, it will sting horribly. Sit on your hands, take up knitting, clap very loudly, anything other than touch the area.
I used a children's ice pack (a Boo-Boo Buddy - go ahead, laugh at me!) to numb the area before injecting. Lots of people are fine without as it is an extremely fine needle, but I am phobic, and quickly ran out of unused injection sites, so it helped.
If for whatever reason you really do struggle, you can always ask for a prescription on emla cream, a topical anaesthesia. I found it a bit of a faff to use, but if you have the patience it may help.
As mentioned above, never go too close to the belly button, but use as large an area of fat as you can grab, alternate sides, and avoid moles and visible veins.
The midwife or HV should take the sharps bin.
It may not be pleasant, but the routine use of clexane after c-sections has reduced the incidence of DVTs.
Best of luck with your new baby :)