We need to open Payday 2's local files through steam. Right click on it in your library, then go Properties>Local Files>Browse... . Keep this folder open for later.
Next, go to this site and download where it says "Latest Release DLL". This is what actually mods the game and creates a "mods" folder for you to easily drop your mod folders into. Leave this in your downloads folder as we'll be downloading one more thing.
Now go to this site and download BeardLib. This is a mod that keeps other mods up to date, and is sometimes a requirement for other mods to function properly. Same as before, leave this in your downloads folder for now.
Now that everything's downloaded, open (or extract) the SuperBLT folder. Take the WSOCK file (the only one in there) and place it into your Payday2 folder (you wont have some of the components in the screenshot, such as the mods folder, until after the following step, so don't worry if my folder looks different from yours).
With the WSOCK file in the Payday2 folder, simply start Payday2. Upon doing so, you should be prompted to download the basemod. Select yes, and once it notifies you that it is done, SuperBLT is sucessfully downloaded.
Going back to your Payday2 folder, you should now see a "mods" folder. Open this, and go back to your downloads folder. Same as with SuperBLT, go to your BeardLib folder and open/extract the folder from inside and place it in the mods folder. Upon launching the game, your Payday2 should now be ready to add more mods, and you can see the BeardLib menu from the main menu. That is where you can update mods when BeardLib detects they are outdated.
In order to add more mods, such as custom HUDS, weapons, and heists, all you need to do is place the mods' folder from the download into the "mods" folder in your Payday2 folder. I recommend creating a desktop shortcut to your "mods" folder for easier access (REMINDER: make sure you take any files out of the zipped file that any other mod will usually download in). Note that some mods need to be placed in an overrides folder, so always check the mod page.