You should definitely not try microchipping yourself! First, it can be painful because it's a big needle, and you don't want your inexperience to make it more painful than necessary. Second, the chips can pop right back out (and you might not even notice), if done incorrectly. Our rescue chips dozens of dogs and buys chips in bulk to hand them off to the vet to insert because we don't want to screw them up!
FWIW, I would recommend that you price-check with your local vet clinic before booking the appointment because I've learned recently that some local vets are charging about $100 to chip a dog -- which is ridiculous. They're paying $10-$25 for the chip, so the rest is just profit. Most vets charge a reasonable insertion fee over the cost of the chip, so it's usually well under $50 (including activation -- and check that, as some clinics are putting in chips that require you to pay extra to activate them when you go online to register them).
If you find your local vets are on the "expensive" side for this service, either use the shelter's chipping program, or go to a vaccine clinic in a Petco/PetSmart/Tractor Supply store where a vet will do it very reasonably. This one charges $19 at their in-store clinics, staffed by licensed veterinarians:
Something like half or more of microchips never get registered, and even more aren't updated with valid contact info, so please make sure you don't forget that step, if the vet or shelter doesn't do it for you!