strong
heavy
empty
small
burlap (AmE), canvas, cloth, leather, mesh (esp. AmE), paper, plastic, polythene (BrE), string
brown (esp. AmE)
He brought his lunch every day in a brown bag.

carrier (BrE), grocery (esp. AmE), shopping
freezer, Ziploc™ (AmE)
bin (BrE), dustbin (BrE), garbage (esp. AmE), rubbish (BrE), trash (AmE)
clutch (esp. BrE), shoulder, tote (AmE)
Her crocodile skin shoulder bag matched her shoes.

drawstring, duffel
evening
carry-on (esp. AmE)
New airline regulations banned scissors in carry-on bags.

overnight, travel, travelling (BrE)
make-up, sponge (BrE), toilet (BrE), toiletry (AmE), wash (usually washbag)
changing (BrE), diaper (AmE)
a baby changing bag

book, school
beach, camera, golf, gym, sports
messenger (esp. AmE)
a cyclist with his laptop in a messenger bag slung across his chest

kit (usually kitbag)
medical
lunch (esp. AmE), sandwich
crisp (BrE)
body
The dead soldiers were put on the plane in body bags.

barf (AmE), sick (esp. BrE) (= a paper bag for a person to vomit into on a plane, boat, etc.)
doggy (= for taking uneaten food home from a restaurant)
gift, goody (= a bag given as a gift with a variety of things in it)
We're giving away a free goody bag with every children's meal.

sleeping
bean (usually beanbag)
saddle (usually saddlebag)
money
He could not convince those who held the money bags that his idea was viable. (figurative)

diplomatic (= an official government container that may not be opened by customs officials)
mail (usually mailbag), post (usually postbag)

open, unzip
close, zip up
fill, pack
empty, unpack
cram sth in/into, push sth in/into, put sth in/into, shove sth in/into, slip sth in/into, stuff sth in/into
The camera caught him slipping a CD into his bag.

draw sth out of, produce sth from, pull sth from/out of, take sth from/out of
delve into (esp. BrE), dive into, fumble in, reach into, rummage in
I rummaged in my bag for a pen.

clutch, hold
carry, drag, haul, lug
I had to lug my bags up the stairs.

heave, hoist, lift
shoulder
He shouldered his bag and left.

fling, sling, throw, toss
He tossed his bag onto an empty seat.

drop, dump, put down
deposit, drop off, leave
We dropped our bags off at the hotel and went straight out.

check (AmE), check in (BrE)
You can check bags of up to 70 pounds for free.

grab, snatch
She grabbed her bag and ran out of the door.
Two youths snatched her bag as she was walking home.

swing
He was walking along swinging his school bag.

load into sth, load onto sth
She helped me load my bags into the car.

unload
look in, search
gather
They were gathering their bags, preparing to leave.

retrieve
I opened the trunk of the car to retrieve my bags.

bulge (with sth)
The bag bulged with papers and letters.

contain sth, hold sth
be crammed with sth, be full of sth, be stuffed with sth
dangle, hang, swing
She had a heavy bag swinging from each hand.

in a/the bag
The mushrooms are sealed in a bag for freshness.

inside a/the bag
bag of
a bag of groceries

the contents of a bag
The customs officer asked him to empty out the contents of his bag.

have your bags packed
You need to have your bags packed and be ready to go by six.

sling your bag over your shoulder
She stepped down off the bus with her bag slung over her shoulder.

Bag is used with these nouns as the object:
grocery