Where can I change my clothes at the airport?

If you're willing to pay a bit to avoid hassle, Frankfurt Airport has 5 shower facilities scattered throughout its terminals that can be rented for 6 euros. The stalls aren't exactly luxurious, but they're fully separated, private and kept very clean, so you can change in comfort and take a shower if you feel like it.

Most larger airports worldwide offer similar facilities, usually at much lower prices than full lounge access, occasionally even for free.

JPatokal gives the answer for FRA, but in every restroom in every US airport there is a handicapped stall, where it is easy to change, and typically two clothing hooks to help you.

A few suggestions of various locations which might work depending on your situation and locale:

Find a local gym or hotel or any other place that may have a changing room or private space, and ask if you may change in a private area at their facilities. Being polite and asking nicely will go a long way.

For gyms you may have to pay for a day pass, but this is typically inexpensive, and it would give you something to keep busy while waiting for your flight if there's a long wait between activities.

I am aware that there are bathroom stalls but changing pants in there involves some acrobatics I'd rather avoid.

If you can't find other accommodations, bathrooms are fine, so long as you don't forget your towel. Pack a towel that you don't mind getting dirty, along with a plastic bag to keep it in once you've used it. Changing won't require gymnastics if you have a clean place to stand.

If you're driving to the airport, you can change in your car in the parking lot, although this will typically require being a contortionist rather than an acrobat.

Call the airport you plan on travelling from and ask their customer service what they recommend. They may be able to assist you or have advice that would help prevent you from accidentally being labelled a terrorist which would likely land you in a private room with one-way glass.

I've changed in and out of suits in airport restrooms, coffee shop restrooms, in the airplane restroom and in the front, passenger, seat of a 1970s Austin America (r.o.w.: Marina?) while someone else was driving.

Focus. Change shirts, skirts, pants and dresses, and anything under them, in a stall. Stand on your comfy shoes, watch out for wet floors. Look before you commit. Enough paper towels can be carried into the stall and laid on the floor if its icky. Standing at the washroom sinks or other semi-private place risks being immodest, in the opinion of other travelers. But if no-one is there...

Jackets, coats, ties, scarves, hats and shoes can be changed without privacy, socks, if you're quick. As several have noted, formal shoes should go on last, come off first. Carry two plastic bags, so the nice shoes don't scuff each other. Put one in a bag, put that bag and the other shoe in the other bag, at the first opportunity. Switch to comfy socks as you put on each comfy shoe, don't make a show of your bare feet or the socks you took off. Don't change socks at your seat on the plane unless no-one else is seated with you.

Keep comfy clothes, some plastic bags and a couple spare hangers in a carry-on suit bag. Make it a small one that's easy to stow near your seat on the plane.

I used to change socks and underwear at the last hour of intercontinental flights. Refreshing! Especially if you have to a full day of work when you arrive. Start your dirty laundry bag or discard as you wish.

If you happen to be at Sydney airport they have free showers. Plenty of space to get changed.

I have changed clothing a few couple of times in airport bathrooms; as other said, just stand in top of your sneakers/whatever footwear you are using as formal shoes tend to be more delicate.

Another alternative is carrying your suit with you in a suit bag, and changing either in the toilets of the plane before landing if in domestic flights. In very long flights, however, typically the toilet stalls in the airport will be more clean.