Now that the athletics and acrobatics skills have disappeared, is there a way to run faster or to jump higher in Skyrim?
There aren't very many (if any) ways to increase your speed permanently or in a traditional manner: looks like Bethesda scrapped that entirely. However, there are a few non-convential and temporary means to increase it:
The basic way to increase your speed is to sprint. Since sprinting uses stamina, increasing your stamina, as Arkive mentions, or your stamina regeneration rate will allow you to move faster for longer periods of time:
You can also sprint with your horse, although it doesn't appear to be any faster than regular sprinting.
If you want to run even faster, contract lycanthropy. In beast form, you sprint much, much faster than normal.
While not strictly allowing you to run faster or leap higher, there are a few shouts that accomplish nearly the same thing:
If you happen to be investing in the Archery skill, you can pick up the Ranger perk, which will increase your movement speed while drawing your bow.
If you exceed your carrying capacity, you'll no longer move at normal speed. So—while not technically a speed increase—if you want to ensure you stay at your speed, you'll want to carry less stuff or increase your carrying capacity:
Additionally, you can negate the weight effects of your armor, thus maximizing your carrying capacity:
If all else fails and you're still overencumbered, riding a horse will negate the effects of overencumberence while mounted.
Finally, wearing heavy armor comes with a hit to speed. You can mitigate this by wearing light armor, taking the Conditioning perk, or, as Hafax mentions, activating the Steed Stone.
You can use the Elemental Fury shout, while sprinting with a dagger, and doing run attacks. The faster attacking makes your running faster. It is quite effective, I tested it myself.