As a general rule, I would not attempt a multi-species hunt for your first DIY trip.
First off, I have found that most areas that have good hunting for one of the species usually do not have good hunting for the other. There are exceptions to this rule, but assuming we're talking general tags, the opportuntity is usually not there. I live in a state that has a 6 week general archery season and a 5 week general rifle season. The deer and elk seasons run concurrently and I always start out the season with elk and mule deer tags in my pocket. That said, I have mule deer spots and elk spots, and they don't really overlap.
For example, I mostly hunt elk in the SW part of the state. The deer numbers are low in the places I hunt elk, to the point that in a few of my spots, I've never even seen a mule deer. Ever. But it is very good elk hunting. Similarly, last fall I finally saw a herd of elk in one of my mule deer hunting spots. Previously, I'd probably spent close to 30 days in that area over the last 4-5 years and had never seen elk during a hunt.
I mentioned "exceptions" and there are two of them. The first is if you draw a good tag for one of the species. There are places with draw hunts here in MT for mule deer, and many of them have good general elk hunting to boot. If you draw a tag for one of the species, you could very easily have the chance to fill a general tag for the other species. The problem is, if you draw a good tag, do you want to waste time hunting/packing out an elk when precious hours on your mule deer hunt are ticking away? I don't.
The other exception is units with small or average animals of one of the species. I mentioned elk hunting in SW MT. Sure, I do see mule deer in some of my spots. But they're always very small (2x2 or 3x3) bucks. If I was hunting in another state, there is no way I'd cough up nonres license prices to bring home a 2x2 or 3x3 mule deer.
I've hunted 16 seasons in MT and have only ever killed elk and mule deer on the same hunt once. And the elk was a cow. As I said, if anyone has the opportunity to do so, it is us MT guys. Still, I and most people I know hunt deer and elk in areas dozens if not hundreds of miles apart. It is very possible to have a multi-species hunt, but the odds of filling the "other" tag are pretty low in most areas.
You could split the difference and hunt a unit that has so-so hunting for both species, but again, your odds of success drop.
Not to mention, I hunt elk and mule deer differently, and a multi-species hunt is not so simple as hiking through the woods and shooting the first species you come across. Having two tags in your pocket is very distracting and I end up bouncing around from critter to critter, never really hunting one species as thoroughly as I should. Now, I try to focus on one critter per hunt.