After pulling the dryer out and unplugging it, I started by vacuuming the vents on the back panel.
Then I went outside, removed the vent cap and vacuumed inside the duct/pipe as far as the vacuum hose would reach. After that was done I went back into the house and vacuumed it from the inside. Later I switched the Shop Vac hose to the blow setting to blow any remaining lint to the outside and then went back out and vacuumed once more.
I started the dryer up again but it still had the burnt smell so I decided to take the back panel off and see if there was more lint build up.
I also removed the whole lint tray and gave it a thorough cleaning.
I put it back together and now I have a dryer working again! There is satisfaction in being able to detect and figure out solutions for your problems. So, yeah, I'm happy. :)