There are three main parking options at the Field Museum: street parking on Solidarity Drive (the road between Shedd Aquarium and Adler Planetarium), surface parking in the lot just east of the Field Museum, and the Soldier Field parking garage. Street parking is the cheapest, but fills up quickly, surface parking is the closest, and the Soldier Field garages should almost always be available, but involves a decent walk.
Everything is thrown off if there's a large event. On a Sunday in the fall, check for home Bears games. Concerts or other events nearby (typically at Soldier Field or Northerly Island) can also disrupt parking. Shedd, which is next to the Field Museum, has an excellent Traffic Alerts page that they update with major events that might impact travel or parking (both availability and cost).
I decide which parking to try based on the status of the surface parking lot: as I approach the intersection where McFetridge drive ends, I look to the surface parking lot on the left. If it's less than half full, there's probably street parking available. Turn left at the intersection, and right onto Solidarity Drive, and look for a spot to parallel park. If the lot is above half full but still has plenty of space, go for that lot: take a left at the intersection, and take the first left available to loop back around to the lot. If the lot is filling up, take a right off McFetridge towards the Soldier Field parking.
Street parking
Street parking is the cheapest at $2 per hour and is often available if you arrive between 9 and 10. There are several pay machines you can use a credit card with; take the receipt and put it on the dash of your car. Alternatively, you can use the ParkChicago app which only charges a service fee (35 cents) if you stay less than 2 hours; that app uses your car's license plate number to register it. You can extend your parking time from within the app if you decide to stay longer (although the Field Museum doesn't always have good reception, particularly at the lowest level). To get there, go on McFetridge Drive until it ends, turn left towards Shedd, and then continue around to the right where parking will be on the left and right. Be careful about permanent signs marking where bus stops are, and paper signs on the trees that may indicate no parking on certain days (the signs will often be posted for several days, so if you see several others parking you probably can, too, just double check the dates on the papers as you get out). If you don't find parking before you get to Adler planetarium, follow the road around to the left and look for street parking on the way back towards Shedd.
Surface Lot
The surface lot is typically $25, no matter how long you stay (up to 12 hours). You'll pay with a credit card or cash as you go in. To get there, go straight on McFetridge Drive until it ends. The lot is directly on your left, but you'll need to turn left, go past the lot, and then loop around to your left to come back to the lot. Sometimes you can do this just after the lot, and other times that road is blocked off and you have to go a little ways before you can cross to the drive heading the other way to bring you back to the lot.
Soldier Field Garage
The Soldier Field garage costs $20 for up to 4 hours and $25 for up to 12 hours (there are several garages, but you'll use the North Garage between Soldier Field and the Field Museum). You'll take a ticket when you enter and then pay via credit card at a machine just before returning to your car. Go straight in on McFetridge Drive until it ends, and take a right. The garage entrance is your first turn to the right after that. Carefully note the floor (number) and section (letter) where you parked, and it helps to take a look around to get a general sense of your surroundings because even the sections are huge. It's generally easier to take the elevators for the North exit, which brings you out to a small lobby where you'll pay before you leave. Go straight out of the lobby, and you can see the main entrance to the Field Museum. If you need the accessible entrance (without stairs), turn right at the bottom of the Field Museum stairs and walk around the Field Museum that way; the accessible entrance is on the east side..
When there are concerts nearby, parking will switch to a flat fee of $25 early in the day, and a flat fee of $50 about 6 hours before the concert begins (so the switch often happens around noon). On those days, you'll pay an attendant as you go into the garage. See the Shedd Aquarium traffic alerts page for a heads up about concerts happening soon (Shedd is next to the Field Museum).