Here I am

Level: 2598 Health: 10111 Strength: 4964 Defense: 4922 Speed: 5028 Intelligence: 4157 AP: 259 Tier: 13 (max!)

I really don't know what to aim for with speed strength and defence. I think Speed should be higher than the other 2 but again I really don't understand why? Health will be around 10.5 to 11K speed I thought 7 to 7.5K strength 6.5K defence 6.5K

Basically I've still yet to defeat a few very high challengers as well as getting extended wins on impossible challengers.

What is the best way of just increasing Speed?

The quickest way of increasing speed is using Exp points. And

You can use battle tokens to buy as many Potion of Fleetness as you can afford.

Battling the extremely hard and impossible challengers will give you a bunch of both.