I have an older light fixture with 3 bulbs - each bulb has 1 white and 1 black wire from it, and it also has the standard grounding wire. The fixture box in the ceiling has 1 white and 1 black (and the grounding wire). Do I bundle the 3 black and 3 white from the fixture and then connect to the corresponding color on the ceiling box?
In a word: Yes.
In a bunch of words: Yes, this is called connecting in parallel.
All household lighting loads and most appliance loads operate at the same voltage. So each lamp, fan, toaster, phone charger, etc. needs to be connected between one of the hot legs and the neutral bar in the service panel.
In the case of your lamp, each bulb needs its own separate connection to a hot (120v) supply and a neutral.
The circuit connection considerations of household wiring are very simple, and can be explained in a page of text. Two or three pages with the diagrams. The bulk of an electrician's work, and the content of the NEC, are concerned with safety.
Speaking of safety, you should use two wire nuts for your job. Different manufacturers use different color codes for the sizes, so you need to look at the chart on the packaging to determine the size to use to join one supply conductor to three flexible cords. There is a lot of leeway here. If you can get the nut onto the wires, then it's big enough, and if you can't pull any wire out of the nutted connection, then it's not too big.
(Personally I prefer the one-word answer but American schools have trained me to keep writing until the proctor says stop.)
Yes, all the whites go together, and all the blacks go together.
Lamps are provided this way for a couple of reasons, first it makes the fixture a bit cheaper, and second, breaking out a separate neutral for each bulb it allows the three lights to be on separate circuits or power sources. Think a 3-unit apartment with a shared hall or porch light, so each tenant can turn their bulb on individually. Or one of the bulbs is generator or low voltage DC power.