They come in various sizes and I re-use the trays from year to year and just buy the boxed pellets.
Take a look at this gizmo for making your own planting "Pellets".
https://www.amazon.com/Carrot-Design-Newspaper-Biodegradable-Seedling/dp/B01DE38236/ref=sr_1_48?ie=UTF8&qid=1533412128&sr=8-48&keywords=seed+planter+machine
You just cut some newspaper or any type of rapidly decomposing paper around the plunger, push it into the wooden mold and you have a very nice small "pot" to fill with your gardening dirt mix. We use ours all the time.
Also, here's one more gardening gizmo that I really like. Its a small plastic shovel shaped seed planter with a "clicker" that you turn with your finger while holding the planter over your seed hole or dirt. A couple clicks and the loaded seed travels down the spout and pops out the end. Much more dependable than trying to shake seeds, one at a time, out of a packet or your fingers. The trick is to just hold the planter in your hand and don't let it touch your other hand or the side of the tray. It has to "jiggle" for it to work.
https://www.amazon.com/LUSTER-Rapitest-Vibrating-Seedmaster-Vegetable/dp/B01LWE4M9B/ref=sr_1_46?ie=UTF8&qid=1533412425&sr=8-46&keywords=seed+planter+machine
I've used it for both large and tiny seeds, works great.
CB from mid-Gulf West Florida...
I don't have any problem starting indoors. They grow strong and full.
The problem happens when I plant the seedlings in the ground.
At that point they just stop. It take a good month for them to start showing any growth. And although I do have a few melons growing, most plants are just now (early Aug) producing massive blossoms. The bees are having a field day.
This doesn't happen with any other seedling.
My research tells me they're 'heavy feeders' and I believe the nutrient I'm missing is nitrogen. I could be wrong so I'll be experimenting with other methods as well.
I'm going to try a couple different things next year such as planting seeds directly into the soil. Some people say you shouldn't do indoor watermelon seedlings because they become root bound. I will also test the soil, something I've never done because everything else has always grown fine.
But I believe the problem is with lack of nitrogen.