More fast
So I decided to take some time during one of my lunch breaks to see just how fast one might go in D&D (without high level spells or flying or stuff). Now, my math might be wrong, and I could be missing some things, but I think I've figured it out.
Wood elf, level 1 druid for the Longstrider spell, level 14 monk for unarmored movement, level 5 elk totem barbarian for fast movement and rage speed boost, mobile feat.
Cast longstrider and then rage. You speed is at... 35+10+25+15+10+10=105 for standard movement. Dash action makes that 210 ft. Double dash with step of the wind makes that 315 ft l. in 6 seconds. If you have a hypothetical item that does small damage to yourself every round, you could sustain this for a minute... total of approx. 3150 ft. That's roughly 35 mph.
I think you could run through (thin) walls at that speed.
Alternatively, drop the level in druid and 1 in monk, pick up two in fighter to get action surge, and be able to go 450 ft. in 6 seconds but only those 6 seconds. (I can't remember if you get another bonus action on a surge, so I didn't include that potential movement).
Like I've said before, the swiftstride shifter race from Eberron is 5 ft. faster than the wood elf while shifted (which lasts for a minute) but you have to fuss more with action economy to figure out the best initiation of that speed. Also I have no clue how canonical that race is. Granted you could increase the swiftstride speed even more if an enemy ends each turn in melee range of you so you can reposition 10 ft. more (I think) as a reaction each turn.
Edit 1: mark of passage human, same as the first... 40+10+25+15+10+10=110 ft in 6 seconds. 220 with dash, 330 double dash. Approx 3300 ft in a minute. Roughly 37.5 mph.
Edit 2: not sure what happened with my math, but that initial action surge number is way off. Fastest action surge would be tabaxi at 425 ft in one turn, second fastest is mark of passage human at 380 ft in one turn.
