I am currently working on some knitted felt coasters. As the name implies, first they are knitted and then they are felted. I have made these many times before, so over the years I have come to favour certain yarns and particular needle sizes. I also have a favourite way to felt.
I use 8 ply, (UK DK, US worsted weight) yarn, 100% wool, not machine washable. I prefer to felt coarse wool which is also cheaper than fine and superfine wools that are used for garments. The needle size I to is prefer is 6.5mm (UK 3, US 10 1/2), this is 2.5mm larger than I would use if knitting the same ply/weight wool for a garment.
The next step is the easiest, Felt by doing the very thing you must never do to your favourite hand-knit woollen clothes – machine wash in hot water on the normal cycle (I usually toss them in the denim load for extra agitation).
You can then either block them as you would any other item. Once the machine is finished, lay them on a blocking board (Google DIY knitting blocking board), stretch and pin into shape. Leave for a day or two, depending on weather, to dry.
Alternatively, (my favourite) toss them in the tumble dryer, then pull it into shape before it cools. If the tumble dryer distorted it too much, simply spray with water and block (as above) or iron into shape. Seriously, you can be as rough as you want.