This is not a trip report.

If you can handle your luggage, you should take the Orange Line from Midway and transfer to the Red Line at Roosevelt. (Orange is an 'L' [elevated] and the Red at Roosevelt is a subway -- transfer is free and easy, just follow the signs.) Take the Red (direction = Howard) to Addison to come out right by Wrigley Field, or, depending on where exactly you're final destination is, use Belmont or Sheridan stations if they are closer. Hail a cab if needed. Here's a zoomable area map showing the stations in the Wrigleyville neighborhood.: http://www.transitchicago.com/travel_information/station.aspx?StopId=17

Thanks - a few followup questions - is it easy to go from the airport terminal to the Orange line with my luggage? Do I need to take elevators,etc. and is the walk long? When I get to Roosevelt Station do I need to go up and down stairs ( again my concern in the luggage) to get to the Red Line? Many thanks again.

Follow the signs in the terminal that say "CTA Trains to City" from the baggage level. There are elevators and escalators.

From Midway. It's a bit of a hike to the orange line from the terminal. I believe you may need to take an escalator, but my memory is hazy. From the orange line, you can transfer to the red line at Roosevelt. The orange line is elevated and you will need to take the elevator down to the street. Then again down to the red line since it's a subway. The Roosevelt stop tends to be busier. You can take the red line to Belmont, Addison,or Sheridan.

From O'hare. You can take the blue line to the Addison stop and hop the #152 Addison bus. It can be a pain in the ass if the bus is crowded, especially with a heavy piece of luggage. You could always grab a cab right on Addison. There is also a Belmont blue line, but that is a subway and would require you to take an elevator to the level off the platform and then an escaltor up to the street....or drag heavy luggage up a ton of stairs. There is a Belmont bus stop (#77) right outside the el station. Cabs are more difficult to catch here in the evenings and later on.

IMO....it's easier to get to Wrigleyville from O'hare.
Taxis cost a ton.
There is also a van sharing that can take you into the city from both airports. I can't think of the name.....if I remember it, I'll post it.

It's a fairly long walk to the CTA at Midway, but if I recall correctly, just one set of stairs/escalators. (I may be misremembering, but I think that's right.) It's well marked. I'm not sure about the Roosevelt station.

The walk at Midway is long enough that you wouldn't want to be dragging two heavy bags, even if one does have wheels

I always take the trains from both airports but never with heavy luggage. Take a taxi from cheapest flight.

The other alternative is to take the Orange line to downtown and get a taxi there to Wrigleyville. Of course if the walk from the Southwest Terminal to the CTA is very long then it may be worth to take a cab.

I fly SW to Midway. It is a walk, but it is not that long, it is flat, there is an elevator. If you are able to walk around town, I don't think it is a problem. It is so convenient and so cheap to take the train into town. You can then take a cab or the bus if you don't want to change trains. There are a lot of people with suitcases walking around Chicago.

Why are you taking such a heavy suitcase and carryon? Are you staying a really long time. When travelling, I love the lighter suitcases. There are some great deals on ebags. You can probably purchase a lighter bag for what you would pay for cab fare--or don't bring the carryon. Pulling one suitcase should be doable.