Generally speaking, installing a new chain on a bike with old sprockets will lead to skipping. The reason is the old chain has 'stretched' and worn the cogs to match. This wear on the cogs is not visible to the naked eye.

For top performance, you should replace the cassette and chainrnigs (esp. if they are visibly worn) when you remove an old chain.

You can tell if your chain is worn by measuring the length of 24 links (12 sets of inner and outer plates). An unworn chain will be exactly 12 inches from centre of pin to centre of pin. A work chain is 12-1/16 or more.