I just received a pair of Ultegra brakes to replace my current setup on my road bike. I've never done this before, and I can't find an adequate guide online. Do I just disconnect the brakes, remove the old ones, slap the new ones on, stick cable back in? Do I need to replace the brake cables or can I use my current ones.

Also, how do I pull the brake cable out of the brake since there's a stopper at the end of the cable?

Does this help? https://www.parktool.com/repair/readhowto.asp?id=20

You can change the brake cable, but it may not be necessary.
The stopper at the end can be removed with a pair of pliers just uncrimp it. If you are careful enough you can reuse it when you get the new brakes installed. Or you can buy new ones at the LBS for almost nothing.

I just did this on my wife's bike today. Biggest pain in my ass ever.... and I even know what I'm doing. You'd think you just unbolt the old brake and cable and then throw the other one on. Not always the case as I figured out today.

That's what I thought. Until my cables frayed, and I realized those needed replacing to get anything done. Which opened a whole new bag of trouble.

Good grief. Cable inners also cost next to nothing, and should probably be replaced if they've been around long enough to wear out a pair of calipers. They are generally a cinch to install, especially if you're doing the calipers anyway.

Mechanic work is not for everyone. If you're not 100% sure of what you've done, please get someone to check your work for you.

The hardest part of installing brakes is cutting the new cable when you're done if you don't have a cable/housing cutter. Get a decent one. If you have a few bikes you'll use it more than you think and you won't wait so long to replace worn or frayed cables.