You shouldn't need to take your shirt off. Unless you are meeting with a professional seamstress who is going to be making custom fitted corsets for you.
A typical bra fitting is done in a retail store, often in or near the dressing rooms. You should wear a bra that you currently own, and a tee-shirt.
Typically the fitter has been trained to measure from the back, so their hands don't brush your breasts, embarrassing you or her.
They will usually hand you the tape to help it get around your body and back into her hands on the other side.
They will need two measurements.
The ribcage/band size: The tape is wrapped around your rib cage directly under your breasts, snuggly, but not tight. You should breathe normally. The assistant may keep one finger under the tape measure to ensure it is not too tight.
The bust/cup size: The tape is wrapped around the fullest part of your bust/breast tissue. Again it should be snug, but not tight, or cutting in at all.
The ribcage measurement is equal to your "band size". When someone says their bra size is 34B, it is the 34 part of the size.
The Cup size is the difference between the "bust size" and the "ribcage". The number of inches difference equates to a letter. So if the measurement around your breasts is 1 inch more than around your ribs that is equal to an "A" cup.
1" greater than ribs = A cup
2" greater than ribs = B cup
3" greater than ribs = C cup
4" greater than ribs = D cup
5" greater than ribs = DD/ E cup
Also note that the cup diameter changes slightly as the band size gets greater. So the "D" cup for a 44D is in fact larger than the "D" cup for a 36D.
There are also specialty bra sizes in sizes above A, and beyond DD. Typically band sizes are available from 32 inches to 42 inches, but again specialty sizes are widely available in stores that specialize in petite and plus sizes.
Your question was perfectly reasonable, don't be embarrassed.
Also wearing a bra or not does not affect the amount of "sag" your breasts will develop unless you are athletic. In that case it is important to wear an appropriate "sports" bra. They look like the top half of a swimsuit with a "racer back" And are often quite snug. If you jump up and down and your breasts jiggle the move up and down too, the bra isn't doing its job. Keep looking.
Other than that the only things that affect the sag, is genetics, pregnancy, and hormones.
I hope this info helps.