I average 37 at 4x4, 1:08 at 5x5, 2:25 at 6x6 and 3:45-3:50 at 7x7, and would say that I am ok at big cubes, so I may be able to help you
Tl

r: It depends on the type cube. Just doing solves will help you, and practicing efficiency will help you in the long run. These are specifics for each cube, but they can blend together depending on method/size of cube/stage of solving.
4x4: Basically just solves to improve efficiency for Cross edges and L8E for Yau (You should 100% be using Yau at this point). Also you want to be around 10 at 3x3 to be sub 40 (meaning you should have sub 30 Yauduction) so get good at no inspection 3x3 as well to get at 3x3 stage. Since it appears that you are this fast at 3x3, most of the time you should be cutting right now will be cut by efficiently doing cross edges and L8E.
5x5:
Basically cementing which method you want to use and focusing on getting really good at it. There is Yau and Redux, and you should try both so you can have a fair chance to choose for yourself.
If you chose Redux: Using inspection wisely to pick your first center, learning shapes of centers and knowing how to recognize them, and having good L12E by hyperfocusing on look ahead.
If you chose Yau: Efficient cross edges are a must, figuring how to get good at fingertricking centers is a must, and having good l8E by hyperfocusing on look ahead
If you chose either method: Be aggressive in 3x3 stage: Your brain tends to have the worst solving of 3x3 stage of any big cube.
6x6: Basically start switching from line forming primarily (1x4’s) to block building (1x2’s, 2x2’s 2x4’s, 3x4’s). Centers is where you can sooooo much time by getting better at intuitively making blocks. PRACTICE DOING SOLVES LIKE YOU HAVE TO EXPLAIN EVERY MOVE YOU MAKE. Force your brain to stop just making lines (I can be guilty of this at times) and start making different shapes. You can integrate this into regular solves by starting out doing this just for the first centers and the last centers (where it is easiest to have good lookahead), then integrating in the entire solve.
7x7: Literally The same as 6x6, but also EFFICIENT EDGE PAIRING FTW