One of my first investments was a 20” Stihl chain saw. Starting with little experience, about 10 chains, one sprocket and 2 replacement bars later I'm finally getting pretty good at felling trees. A 20” inch bar is a good size for using with a sawmill, as it can fell trees up to 2-3 ft in diameter. Having a smaller 12-16” backup saw will be a lifesaver the first few times your bar gets stuck in a tree. It's also much lighter and easier on your back for small jobs.
One thing to consider is the cost of chains. I get my 20” pro chains locally for about $15 each, but most places charge over $22. At this point, accessories and replacement parts have cost as much as the chainsaw, so plan accordingly.