Planning a trip to Yellowstone for May 2017( the 17th-21st ). We are planing on driving from Denver to Jackson, WY. Looking for input about staying outside the park. We will be departing from Bozeman Airport on our final day so we figured we would start at the south entrance and work our way north. Is it practical to move around from Jackson to West yellowstone and then spend our last night in Gardiner, MT ( we want to book a night at Dreamcatcher tipi resort)....???

See if you can score a room inside the park at Mammoth.
Otherwise, something in Gardiner is OK but not as good as Mammoth IMO.

You are much better off in the Park. West yellowstone has good hotels, but it is a haul off the main loop. did you look for lodging in the park? Even with kids who like pools, we stayed in yellowstone.

West Yellowstone is fine, I like the drive from there to Madison Junction, not long at all. Gardiner is also scenic, and minutes from Mammoth. The lodging options are simply better, more varied and cheaper, outside the park and the Xanterra stranglehold. (I speak as one who worked in the area and has visited for 40 years.)

I agree with the first two comments.

It is much, much better to stay in the park. I would recommend a split of two different in park lodges - maybe something in the Old Faithful area and then something in Mammoth or Roosevelt. I am just speaking for myself, but the advantage of the lodging in the park is that it allows you to be THERE after the day crowds leave. You can't beat waking up at the Old Faithful lodge and walking through the geyser basin early, before the crowds arrive. Same goes for the evening time. Also, it is dangerous to drive after dark and staying at various places throughout the park allows you to not do that. Staying outside the park adds a lot of driving time and you will do a lot anyway, just driving within the park.

Or you could just wake up early; really, the distances to cover are minor (20 miles from W. Yellowstone to Madison Junction, 30 to Old Faithful; if you are in Gardiner, you are less than 1 mile from the park entrance, and ~5 mi to Mammoth!), and in the early morning there's a lot of wildlife to see as you drive the routes.
I've stayed outside the park (all of the entrances) and in the park (Old F. Lodge, Yellowstone Lake (Cabins), and Mammoth), and speak from experience of both. The scenery and wildlife don't stop at the park boundaries, after all.
And my statement about the price, etc., and Xanterra hold true (and what about those who can't get reservations for those overprice digs?).

I also noted that the OP said specifically "Looking for input about staying outside the park" not lectures about how to visit or not visit.

I usually like to stay inside national parks as well but Gardiner works fine as an alternative to Mammoth. We've based in Gardiner twice and the drive to Mammoth is very short. There are a lot more dining options and the grocery store in Gardiner is better stocked vs Mammoth (although for one night it probably doesn't matter). Yellowstone/Tetons cover a large area so it's practical to base in different locations. This past May we stayed in Gardiner, Old Faithful and Jackson over the course of a week.

A poster on this thread has brought up cost twice now. That's fine but not everyone makes their accommodation choices solely on what is the cheapest option. Some people (myself included) are willing to pay more for prime location.

Lastly, there are cabins in the park available for under 100$/night. Not expensive. Not overpriced - you pay for the location. Worth every penny.

The tipi the OP is considering is over $300 a night.

Not all of the in park accommodations are open at that time of year. For inside the park, I might choose between Old Faithful and Lake Yellowstone (Roosevelt is not open).

how do the phrases "better, more varied and cheaper" and "what about those who can't get reservations for those overprice digs?" translate to "solely on what is the cheapest option"
That is not only misinterpreting what I posted, but deliberately insulting.
There are so many ways to see Yellowstone; and not everyone who wants to can stay inside the park, hence the proliferation of lodging at and near all the entrances to the park.