Ingress level requirements start out pretty easy, but levels 9-16 seem to have some confusion around them. You need something called AP to advance, at low levels good sources for AP include hacking enemy portals but as you level up the best methods are capturing portals, linking, and fielding. In portal-dense areas you may be able to level quickly through destroying enemy portals, but if the portal density is more sparse you'll spend a lot more time attacking and capturing the portals will give you the bulk of the AP.
Starting at L9 you'll need badges. Badges are a little tough to figure out, a gold badge counts for a bronze and silver badge as well. Of course, a bronze badge counts for nothing... For Level 9 you need 2.4M AP plus you need 4 silver badges and 1 gold badge. But since the gold badge also counts for a silver you only need one gold and three additional silver badges. This becomes the most important with L12, here you need 7 silver and 6 gold, it's important to understand that this is only 7 total badges and not 13.