In this Instructable I'm cutting an orange flesh melon in most of the pictures, though a cantaloupe also appears. This technique is pretty much universal, though. As a bonus, you don't really even need a cutting board to do it this way - perfect for picnics, and for avoiding invisible onion or garlic contamination on your cutting board.
Check out the video to see just how quick and easy this is:
Step 1: Choose Your Melon
First, make sure it's actually melon season in your area, as imported melons will almost certainly be sub-par. Melons like hot weather, so melon season usually starts mid-summer, and continues through early fall, though times may vary in your region.
Second, make sure the melon smells good! Give it a sniff: it should smell floral and melony, particularly at the end where the flower bud used to be (the spot opposite the stem end.)
Third, give it a light shake. In cantaloupes and some types of honeydew, the seeds loosen up enough as the melons ripen that they'll rattle around a bit inside when the melon is ready to eat.
If the melon isn't quite at peak ripeness, leave it on the counter so it can ripen a bit more, then put it in the fridge to arrest further ripening. (This is only going to help so much; a completely green melon shipped across the world will never truly ripen.) Bonus: in hot weather, fridging the melons means they're nice and cold when you're ready to eat them. The perfect summer snack!
Step 2: Cut in Half
Be sure to wash your melon thoroughly before cutting, so you don't carry any surface bacteria into the edible melon flesh. (Thanks to thematthatter for the reminder!)
Step 3: Scoop Out Seeds
Your fingers are better-suited to seed scraping: they get in all the nooks and crannies nicely, and you can separate the seeds without gouging the soft ripe melon.
Use a bit of fingernail action (or make a small cut with your paring knife) to detach the seeds from the stem-end, and scrape the seeds out using your thumb. You'll quickly see where the seeds are attached; run your thumb nail-first along these attachment points, and tilt the melon to smoothly scoop the seeds out.
Step 4: Slice Into Wedges
Cut your half in half, then cut each quarter into 2 or 3 roughly equal wedges.
Step 5: Remove Rind
Just hold the melon wedge in your hand, and slide the paring knife between the rind and melon flesh along the line where the color changes.
Advanced technique: I often leave the last little bit of melon attached to the rind. This keep the melon wedge from slipping off its rind as I slice it, which is super-handy. Check out the video in the intro step if you're not sure what I mean.
Step 6: Slice!
Yes, of course you can use a cutting board; but this method is much faster, and easier once you're used to it.
Step 7: Serve
I like to squeeze a bit of lime juice over my chopped melon, as it brings out and deepens the flavors. Give it a try, and tell me what you like best!