I'm going to assume that you're not very familiar with roses in general. Please forgive me if that's incorrect, but I'm doing so based on how your question was written, and because I can't find any other posts you've written here. So if I'm over-explaining something, keep in mind that I'm just trying to be sure I'm being clear.
:-)
OK, so looking at the pic, I already see some dead twiggy ends. You'll probably recognize that they should go. But look at that as a sign, and instead of just snipping it off, get down on the ground and look at it. Trace a dead twiggy end down the stem until you see what looks like a "replacement" shoot. There will be an intermediate zone between the dead and the replacement, and in that intermediate zone, the stem will look "wimpy" or "worn-out" in comparison to the replacement. You may see multiple shoots at each growth point -- normally, only the central shoot should be growing, and if you see two or three, it means that something went wrong, and the central bud was no longer able to keep the two side buds asleep.
Basically, for whatever reason -- damage, disease, etc. -- the stem leading to the dead twiggy ends was on its way out, and being on the way out, was no longer able to keep quiet a bud lower down, which awoke and became that replacement cane. You'll notice that the replacement is taut, or "inflated" in appearance, compared to the "on its way out" segment above it. You'll notice that it's the only shoot at that point -- you won't see two or three coming out there. It'll seem "further along" than shoots immediately above it. Get familiar with those signs, because I'm pretty sure you'll find them in abundance as you work your way through the rose. Now, beginning from the base at the ground, trace stems upward until you get to a strong new replacement shoot. Look at what's above it, and if it matches what you saw before that led to the dead and scraggly ends, make your cut. Start there -- remove all those "on its way out" stems by cutting just a half inch or so above that strong new replacement shoot. This should remove a log of scraggly bulk by repeating one basic step throughout the rose.
Once that's all done, look at what's left. You'll notice that new shoots grow in the direction they're pointing from the stem. If you cut back anything else after removing the "on its way out" stuff, do it so that the top shoot is pointing away from the center of the plant, and try to avoid having two nearby stems be cut back so that their shoots will grow into each other, or cross and rub. Repeat through the plant.
Now, you're basically done. Just go over to make sure you didn't miss any dead nubs at the base of the plant, clear any weeds that might have taken residence at the crown, and neaten up the mulch again.
Good luck!
:-)
~Christopher
