I started with a size 8, which is trickier to insert because it's so flexible, but it's much more comfortable when you're starting out. I use an 18 now because it's easy to insert because it's more rigid, and it drains much faster. But I've been cathing for years.
I still find it uncomfortable to use one where I have to apply a lubricating jelly separately. Hydrophilic catheters have a water packet inside their package, and you squeeze it to pop it and wet the catheter before you even open its sterile wrapper. This activates a lubricating layer on the outside of the catheter, which makes the whole thing very slippery. The lubricating layer is also attached to the catheter, so it doesn't just rub off in the urethra the way Surgilube and other separately-applied do, so it's lubricated so the way in. Finally, I use a Coude tip, which goes through the final sphincter much more easily. I like the Lofric Origo brand, which uses saline that activates the lube more quickly than plain water, and has plenty of liquid in the package so I don't end up with dry spots where the lubricant isn't activated.
Hydrophilic catheters are more expensive, but my insurance covers them. I find pain PVC catheters to be a little barbaric, especially when they first then onto new users.
All your urologist if you can get some samples. My doc has a closet full of catheters, some supplied by manufacturers and some by patients who end up with more than they need. I take her a case when I have some left over, since I don't always have to cath 7-10 times a day like I am at the moment (insurance provider me 240 a month, which is 8 a day).
Also, ask about getting Urojets. These are disposable syringes filled with Lidocaine gel, with a smooth plastic tip intended to squirt it in the urethra. It's an easy way to get numbing gel in there, and helps when my urethra gets irritated. Again, my insurance pays for these, and my doc was able to get me same samples before she wrote the prescription. I had to show one to my pharmacist so he could order the right thing (they'd never heard of them) but now it's just a routine prescription that I pick up every month.
As for leaking after you cath, that means you're not quite getting all the urine out. I find that leaning forward from the waist helps, whether I'm cathing while sitting on the toilet or standing in front of it. Sitting tends to work best for me, and I just keep a plastic container in the bathroom to put on the floor in front of the toilet to drain the urine into. It's another thing that my urologist have me, and it has markings so I can measure residual volume to keep track of how my bladder is doing.
Finally, I find it helpful to pull the catheter out of my sphincter, then push it back in to make sure there's nothing left in my bladder. If you're already irritated, though, that might not be a good idea. Be sure to pull the catheter out very slowly, and if more urine starts to flow (or even drip) as you withdraw, stop pulling it out until the flow stops.