Besides instantons and large-$N$ what are some other general non-perturbative methods for quantum field theory? Besides large-$N$, instantons, lattice QFT, what are some other non-perturbative methods that help us better understand QFTs like the large distance dynamics of Yang Mills and QCD?
 A: One of the classic non perturbative tools is the method of Schwinger Dyson. The Schwinger Dyson Equations essentially give you an all orders (extremely complicated) implicit integral equation for the propagator / vertex etc of a given theory. Since they are non perturbative they are extremely hard to study and usually one tries to truncate the equations at some fixed order. Doing this consistently is non trivial.
On the context of bound states I feel obliged to mention the Bethe-Salpeter approach. Finally other numerical approaches such as Monte-Carlo simulations or worldline numerics can complement lattice approaches and are inherently non perturbative 
A: Resurgence/transseries/renormalons, Instanton's cousins (Monopoles, Gribov solution...), DS equations (already commented) and lattice.
As far as I know, resurgence is trying to "see" beyond-all-orders effects in finite expansions. Sometimes, you have beyond-all-orders information hidden in perturbative terms. I don't know exactly why this happens, but it's not hard to find material online.
Lattice is only interesting using computational techniques, but reading about it you can get some insight.
DS is well-studied too. In analytical and numerical works.
Topological solutions are my favourite tool. Try Shifman book.
