Most of the time, I split the pelvis. I've used a variety of tools. So far, this is my hands-down favorite:
They're only $7 bucks, and they do an amazing job. It's a fine-toothed pull-saw, so it's much easier to get started. I keep a bunch of them around. It's also the best saw I've found for sawing off the hooves. They're great for pruning too.
The thing about splitting the pelvis is that it goes hand-in-hand with the way I remove the anus. After the cavity is pretty well emptied out, I'm left with the last bit of the alimentary canal hanging over the top of the pelvis. I saw through the pelvis, and fit the intestine through, and now it's all hanging down with the anus as the only thing attached. I take my knife and make a cut to either side of the anus and it all falls off into the gut bin.
Of course, this assumes you're starting, like me, from a head-up configuration. Gravity is doing all the work for you. Honestly, this is one of best reasons I can see for doing a deer head-up. The poop hole is always outside the deer. Anything that comes out goes on the ground.
Care for the saw: Run through the wishdasher or clean in the sink and then give a light coating of vegetable oil.
You're doing just fine for a first impression, as far as I'm concerned. I know I'm a minority opinion on this topic, and the way we clean deer is. . . well, at times it can get we're more tied to how we clean out a deer than we are to the way we take Communion. It turns out my methods are more popular in the South, and it makes sense, since I learned to clean a deer from a guy from Alabama. Guys up North think I'm strange.