I live in a hilly place and when biking uphill. Using the smallest gear on the crank and the cassette, the chain skips over some cogs. I identified it is only on that combination (smallest gear on both).
I suspect I need to change the cassette, but is there any other alternative path I can take?
I assume you mean the easiest gear on the front and back, meaning the small chain ring up front and the large sprocket in the back.
Start with a new chain first. If you get a new cassette you’ll also need a new chain anyways. A bike shop should be able to look at the drive train to see which part needs to be replaced.