You can change your base speed using the console, which you can open using the ~ key (or whatever key is to the left of the number strip at the top of your keyboard).
Type player.setav speedmult
x
where x is the number you want for your walking speed; the default is 100.
You can either increase or decrease this number.
Increasing it higher will result in super fast running which man ruin the reality of the game.
I'd recommend player.setav speedmult 125 .
In my opinion the default running is way too fast, and the default walking is too slow.
While holding Alt to sprint is perfect.
The only other way I know to edit these values is to open the core files, like XML with a text editor, but I won't go into that much detail here.
You should be able to Google it.
gaming.stackexchange.com/users/14693/Morebatteries
Morebatteries' answer works pretty well - but I've also discovered a mod that is a little more nuanced.
Realistic Running Speed
According to the documentation:
It adjusts the speeds of the player, NPCs and horses by the following ammounts:
Character Sneak Walk +15%Character Sneak Run -25%Character Walk
+15%Character Run -20%Character Sprint
-20%Horse
Walk
+5%Horse Run 0%Horse Sprint
+5% After a quick play it appears to be decent.
gaming.stackexchange.com/users/17403/bradhouse
There is unfortunately not a lot of control on PC; you have only full blown sneak running or tip-toed Hall of the Mountain King sneaking.
Getting a wired Xbox controller is a good solution.
gaming.stackexchange.com/users/14291/Skizzlefrits
My Realistic Humanoid Movement Speed plugin fixes everything.
Your walk/run/sprint/sneak/horse move speed will be realistic.
gaming.stackexchange.com/users/53880/No more turtle moves
You can press Caps Lock to permanently run.
The default for sprint is Alt .
gaming.stackexchange.com/users/19799/emil
Hit caps lock and it adjusts the speed between intolerably slow and "still slow as molasses but slightly better".
gaming.stackexchange.com/users/20407/Chris
When I'm sneaking around, I can move more slowly and quietly by holding  shift .   But walking this way is  really  slow. I would like to move a little bit faster, but not full speed.    On a console I imagine you have much finer control over how fast you move with the joysticks. Is there any way to accomplish this on the PC (without an external joystick)?  Perhaps there is a console command or config file setting that will change the % speed modifier that holding down  shift  uses?
Is it possible to adjust your walk speed?
gaming.stackexchange.com
<the-elder-scrolls-5-skyrim>
gaming.stackexchange.com/questions/36069
gaming.stackexchange.com/users/5291/dpatchery
You can change your base speed using the console or get the Realistic running speed plugin.  Walk speed could be adjusted via console command, different controller or certain mods.
It is argued that a console command could be helpful. A different controller could solve the problem. It is suggested that certain mods and plugins could be useful.