Maybe a little too zen, but like most things in life, if you keep putting it out there, eventually the Universe will kick in.
Expect a fish on your next cast so make it a good one and cast to where you think your fly should go, get a good drift over the spot that you think is holding fish. If it doesn't happen, reset to zero, it will on the next one, so cast again. Repeat as necessary... Eventually it will work.
Fish with intention, pick your spots to cast to that seem like they should hold fish- current seams, at the base of riffles, through runs, heads and tails of pools. Give some thought to the fly you choose, and where how you present it. Look at the water for a bit before you wade in and start chucking stuff and try to analyze what's working and what's not. You'll still have to earn your fish, even in Montana!
Flyguy said it very well, expect to have a great time, no matter what the body count is.
As far as improving your odds, I'd do a couple things.
back home, work on your casting accuracy at different points 20-40 feet away as well as some distant ones. Get to the point you can be comfortable, say landing within a hula hoop and work your way out to 60, with some longer casts going for distance where accuracy might be less important (for wind and chucking streamers on big rivers).
try and get some time on moving water, even if it doesn't hold trout to practice mending line and get good drifts.
If you can get a trip or two in for trout before you go for Montana, by all means go for it, but even if you can't try some nymphing, using a dropper or indicator for whatever might be around- bluegills, bass whatever, especially if it's on moving water and try some of the different presentation methods mentioned at the westfly site (see next section).
google up some hatch charts for the rivers you plan to fish and make some notes on the different hatches - what flies and sizes imitate the nymphs, emergers, spinners? What time of the day do the different hatches typically come off ? Where in the stream do these hatches occur (riffles, slow stretches, over gravel or muddy bottoms)? Anything about their behavior that you should know? ( A couple good resources for this, once you get the hatch charts are Troutnut.com Fly Fishing for Trout
and the entomology section of Fly Fishing | Westfly
) At the Westfly site, you'll see recommended patterns, as well as hot links on different methods of presentation. These are helpful to review to know different ways of presenting flies, especially wets and nymphs.
Arrange for a good guide in advance. If you're going to the yellowstone area, you might want to make arrangements through a very reputable place like Blue Ribbon Flies
Discuss in advance what you want to do-- even though your odds of catching might be better in a drift boat you may want to do a wade trip to learn more about presentation, recognizing holding lies, and it will prepare you better for fishing on your own. I think if you emphasize your interest in the learning more than the catching upfront and in getting a guide that can teach as well as fish, it will leave you in the best place to fish on your own down the road. You'll also have a very good shot at catching too.
Consider buying a book with maps (for access points) and tons of info for the major and minor waters in Montana like this one, and start looking through it to find some different places to try:
Amazon.com: The Montana Angling Guide (2nd Edition): Chuck Fothergill, Bob Sterling: Books
(used copy For fifty cents, it also gives a lot of info on YNP)
if you’re feeling more spendy, these are also good:
Amazon.com: Fly Fishing Montana: A No Nonsense Guide to Top Waters (No Nonsense Fly Fishing Guidebooks): Brian Grossenbacher, Jenny Grossenbacher: Books
(new for about 20 bucks)
and for YNP, Craig Mathews book is excellent (he’s the owner of Blue Ribbon Flies) with tons of info on the park's fishing including trailheads etc for the backcountry where you're likely to find less pressured and easier to catch fish.
Amazon.com: The Yellowstone Fly-Fishing Guide: Craig Mathews, Clayton Molinero: Books
Browse through those books to pick out some different water some less pressured smaller streams that might hold some cuts for example, as well as some of the bigger named streams and go over those hatch charts. They can be really helpful in making sure you're in the right place (different sections of the stream) at the right time (morning afternoon evening etc).
In general, you might have the most luck chucking an Elk Hair Caddis in fast mountain freestone streams, as opposed to the more technical smaller fly lighter tippet spring creeks, but mix it up a little with some different water to get the full experience. Plan a bunch of different trips to different water types in advance so you have a bunch of options when you get there.
Once you're in Montana, plan to fish hard, and plan your meals around the fishing-- some exceptional fishing to spinner falls happens in the evening when other folks are having dinner... Plan on coming back from your trip exhausted.
And look up once in a while too-- you'll be in some beautiful country, in the middle of all kinds of wildlife and just soak it all in. You'll have a blast.
Best of luck, and now we're all looking forward to your trip report!