IV starting is a SKILL, and one that will be developed, as another has posted, with PRACTICE.
Most of my experience has been starting IVs on adults.
I'm sure peds IVs take a different kind of finesse (and bless the nurses who are good at these!!).
I can, however, give a few pointers (used to work in a Radiology Dept [X20yr], starting over 20 IVs a day, on all types of veins, for imaging studies):
**Take your time.**
*Try applying the tourniquet and mentally choosing one or two good veins, then let the tourniquet down while you get your supplies set up.
*Or, if you don't have a lot of supplies to set up, apply the tourniquet, ready your supplies, and eyeball the potential veins.
*Either SIT yourself down, with your supplies in easy reach (not tilted on a pillow, so they may fall), or raise the bed to a comfortable height, so you're not stooping or stretching. Put your supplies in easy reach.
*If I'm not sure if I want a particular vein, sometimes wiping it several times w/alcohol makes it stand up/out more (it's not the wiping, I think it's getting the skin wet that makes it look more defined. Just MHO)
*I prefer to use 20g, if the vein is large enough for it. 22g, being smaller and wimpier, requires sl. more handling finesse.
*VERY IMPORTANT to anchor the vein just DISTAL to where you'll stick. Hold the extremity with your nondominant hand, and with your thumb, pull on the skin just below where you intend to stick, to make the vein secure. Do not let up till after the initial stick, flashback and threading of the unit so you know it's well in the vein.
*When you see blood in the flash chamber, advance just a teeny bit more (to engage the cannula, not just the needle tip, in the vein). Then advance the cannula over the needle. This should not cause the pt extreme pain, and you should not meet with resistance. If pain or resistance occur, the IV is probably not in the vein. You may redirect or abort that site and begin again (been there, done that, got GOBS of T-shirts!!).
*Once the cannula is advanced into the vein, THEN let up your traction and use both hands to secure the cannula, flush it or attach the IV tubing.
One day a NICU nurse gave me a helpful hint. When the pt's veins don't pop up, even with the tourniquet properly applied: "I just aim for the blue streak, and I hit it nearly all the time," she said.
Also, I have found that the veins on the inner forearm are the most still and straightest (therefore the EASIEST to hit!), plus when the IV is in and secured, it tends to stay good longer, as it's not on the outer wrist or AC or someplace where it encounters a lot of movement.
That said, those are the veins that often require the "aim for the blue streak" technique.
Start shallow and go mm by mm (sometimes they're shallower than you realize, and you don't want to over-shoot).
With time and practice you'll be skilled enough to go through the skin quickly (for less pain) but then slowly enough to hit the vein without going through it.
Any chance you could shadow the IV nurse for a day or two?
Or shadow another nurse when they have to start IVs, to get their tips/techniques?
Or spend a day or two in Radiology (CT or Nuc Med), where a lot of IVs are started?
Good luck! You CAN do this.