Learning FFT really helped me solve 800's. You can start there.

Start with segment tree.

The list of topics that you should learn for solving 800 rated problems : Li-Chao Segment Tree
Segment Tree Beats
RMQ in O(n)/O(1)
Any self-balancing tree except treap
Link-cut tree
Wavelet tree
Mergesort tree
Binomial heap
Fibonacci heap
Leftist heap
Dominator tree
3-connected components in O(n)
k-th shortest path
Matching in general graph Weighted matching in general graph Preflow-push
MCMF in O(poly(V,E))
Minimum arborescence (directed MST) in O(ElogV)
Suffix tree
Online convex hull in 2D
Convex hull in 3D
Halfplane intersection
Voronoi diagram / Delaunay triangulation
Operation on formal power series (exp, log, sqrt, ...) (I know the general idea of Newton method)
How to actually use generating functions to solve problems
Lagrange Inversion formula

There is a Chinese problemset called ynoi solving it will benefit a lot