To add to what Horace said, the closest CTA station to that Hyatt would differ based on what airport you came from, though they would be close enough together that there wouldn't be much point in changing trains in the Loop. The closest Blue line station (from O'Hare) would be Clark, the closest Orange line station (from MDW) would be State/Lake. Coming off on the Orange line you would have to carry your luggage down some stairs, if I recall correctly. (The Loop trains are on elevated tracks, not underground. the only underground train in that neighborhood is the Red Line.)
This is the CTA loop map, which shows the trains that service the Loop.
http://www.transitchicago.com/maps/maps/F2003D.html The Hyatt Regency Chicago is on E. Wacker between Michigan and Stetson, which is diagonally about four blocks up and right from the upper right corner of the colored Loop area on this map. The traffic in that spot can be rather nasty for pedestrians w/ luggage because it is the confluence of a lot of streets with a lot of traffic, so if you have luggage I would suggest taking a cab from the train the rest of the way to the hotel -- they are easily hailed in this neighborhood.
Your best transit bet pricewise would be to purchase visitor's travelcards in advance from CTA; they will mail them to you. These are good for unlimited rides during your stay on all the CTA trains and on the buses; they are not good on the Metra trains. Technically you can buy them from the airport station vending machines, but I've found that they often run out of them, so I recommend having them mailed in advance; one less hassle at the airport.
Note that there are actually 3 Hyatts in the Downtown/Loop/N. Michigan area. The one Horace and I have mentioned is
the Hyatt, the big one at 151 E. Wacker. If you meant the Park Hyatt (at 800 N. Michigan) or the Hyatt Printer's Row (at 500 S. Dearborn) then the advice would be slightly different.
BTW, the Hyatt on E. Wacker is not close to anywhere where it is simple or cheap to grab snacks/drinks, and they charge bar prices for them. (The hotel has no vending machines for soft drinks.) You may want to do a convenience store run to a White Hen Pantry to pick up a few items for your room.
PS: We take the CTA orange line trains in to this neighborhood from MDW very often, and have never had a problem with anything worse than a somewhat smelly homeless person. The platforms are old in the Loop, and the stairs feel rickety, but the trains are efficient and not unpleasantly dirty. (Don't get the impression that they are spotless, mind, but they are not bad.)