Running 2 Match types
What happens if I run Exact and Phrase at the same time?
If someone types in the Exact keyword, will it show exact,
and if they type in anything else, it will show phrase?
Hi spires, what you've said is pretty much the answer. Adwords always tries to use keywords that are most closely matched to the search term so, as you've said, if the search user types your exact match keyword as a search term, it'll be that term that triggers, if they type anything that matches the phrase, but not the exact, it'll be the phrase that appears.
I should add that this assumes your keywords are running and have appropriate other settings to allow them to display.
There's a whole bunch of stuff about matching keywords here:
Re: Running 2 Match types
It's often useful to bid higher on the exact match, but to also run either a phrase match or modified broad match. Why? You know what you'll be buying with the exact match bid. You don't really know what you might be paying for with the other match type, so bid lower. Also, as you use that secondary match type in conjunction with the primary (exact), you'll identify important keywords via the search term reports. Add those in as exact matches at higher bids as you find them. For example, you might have:
[foo bar] <-- $1 CPC
+foo +bar <-- $0.50 CPC