The cassette will usually wear much more quickly than the chainrings, especially if the chain has a lot of miles or is neglected. A chain that is "stretched" (actually the pins and holes in the links get looser, making the chain longer), the chain rides further up the teeth and wears out the cassette much more quickly. Usually such wear is visibly different on those cogs you use most compared to those seldom used. This is why replacing your chain every 3000 miles or so is so important.

Chainrings usually don't wear like this because there are so many teeth to share the load, but in severe cases it can. More often, the teeth get narrower.

Replace the chain, and cassette if it either shows wear or the chain is obviously "stretched", as one will ruin the other. Replace rings only if obviously toast.

Of course, this assumes your gears are tuned, dérailleur is straight, etc. Otherwise skipping will continue. If skipping continues, it could be your freehub slipping.

I had one bike (ultra stiff aluminum) that tended to skip after taking off from a stop, and never found a cause. I took the same parts and put them on a new frame (compliant steel) and it has never skipped. It can be elusive.