You need to purchase bus tickets from the kiosk inside the airport (or just outside the bus stop at the square if heading to the airport). Bus 384 also goes from the Airport to Mladost 1 metro station, but only runs at certain parts of the day. The bus tickets must be 'stamped' once on the bus to validate them. You do this by putting the ticket into the little metal box and pulling up on the lever to punch holes in the ticket. Maybe every bus has a different pattern of holes to ensure the ticket wasn't used before but I found this is a very rudimentary setup with no date or time stamped on the ticket.
The metro tickets are very different and have barcodes at the bottom. You buy them from the metro station's ticket office or vending machines. Stick the ticket into the barcode reader at the gate with your left hand and walk through when the light goes green. If you feed the ticket on the right-side, someone else might walk through the gate on your ticket. There is no need to scan the ticket when exiting at your destination, but hold onto the ticket during the trip in case there is an inspection. Also note that metro tickets are only valid for the station purchased at and only for a set amount of time. I tried buying an extra one to use later and it didn't work at another station (said barcode was invalid) and also gave the same error message at the originating station the following day, even though I hadn't used it yet... so you can't buy metro tickets for future use, but thankfully you can for bus tickets. I found the metro system very easy to use and there are only two lines so it would be difficult to get lost.
There is a full day pass you can buy for 4 Bulgarian Leva that works on Sofia's metro and buses (otherwise buy seperate tickets for each).