Live in the center! Imagine that straight line between the baskets and go post-to-post, the whole game. Don't veer away from that line unless you have to. Doing so keeps you looking like a gazelle galloping down the court, plus it saves you energy later in the game. I presume there's no screening, but if there is, roll back toward the line pronto.
If 3-to-5-second rules apply so nobody just camps out under the bucket, practice positioning yourself on opposite edges of the paint.
No need to wear yourself out chasing guys out to the 3-point arc and scrambling back to rebound. If you have to come out to defend, don't stray far from the endline. Let your guy have all the mid-range and long-range shots beyond the elbows as he wishes. Preserve your energy for the plays in the paint and around the rim.
When you get the rebound or the defensive stop, find your lead guards, get them the ball and immediately haul butt to the other end. No matter what level of hoops, the smaller guys are always taught to "reward the big guys" with the ball, whenever those guys make the effort to go endline-to-endline with opponents at their backs.
You'll have a little more leeway to roam if the pickup hoops is halfcourt, but still, keep your positioning around the "nail" (center of the free throw line) and the low blocks.
Your perception of speed, as well as the perception of your competitors, is grounded in your relative instinct in getting to the ball. You don't have to be the fastest speed merchant on the court. You just have to be quicker to the hoop and to the ball than your opponent.
Your NBA guys to watch: Rudy Gobert. That guy is a monster right now, and his quickness for a big man is off the charts. Also, Mitch McGary. Much more of that football build you have, and he doesn't look slow at all.
As for drills, here's five from a Men's Fitness piece.http://www.mensfitness.com/training/cardio/5-fat-burning-speed-and-agility-drills