There are quite a few options for moving pasting cutting and copying files to and from your Transformer. This is a mini guide. I am intentionally leaving out a direct connection due to the number of people who have problems with connecting their TF to a computer, though for many people it works just fine. These are alternatives.
Best long term and easiest solution:
Purchase a microSD card, (a regular SD card if you have a dock). If the purpose is to transfer files it doesn't need to be especially big, fast or expensive but make certain it's a brand name purchased from a reputable source. Some microSD cards can be purchased with a USB card reader in a bundle if you don't have a card reader in your computer.
Apps for file management from your computer:
WiFi File Explorer and WiFi File Transfer are only 2 of the apps that will allow you to browse your TF from your computer and will allow you to transfer files in both directions.
File managers on your Transformer:
ES File Explorer and File Manager HD are 2 of my favorites though the default file manager will get the job done. Each has the ability, (along with other file managers like Astro) to browse a share on your computer or network and allows you to cut/copy/paste individual or groups of files, (multi select), in either direction. You must have a share on your computer for this to work. I think ES File Explorer is the easiest, especially if you have a fixed IP address for your computers.
The cloud option:
I have more cloud storage available on my Transformer than I could possibly use, over 65GB. The single most useful way to do this is by getting DropBox. a 2GB account is free, but other free options with various amounts of free storage include, Cubby (from the LogMeIn people), Box.net (BOX on the Google Store), and Google Docs are just a few options. What I like about DropBox s that a stub is installed on my computers and anything I put into the DropBox folder becomes available very quickly on my TF and the same goes for sending files, (through "share") from the TF.
You also may want to check out Sugarsync for something very similar to DropBox.
I have an app on my TF from Wyse Technology called Pocket Cloud Explore. This also requires a small program install on my computers. With Pocket Cloud Explorer I have access to ALL the data on my 3 computers in my office anywhere I have an Internet connection. I can open docs on my desktop computers and save them on my TF.
Hope this makes things a little easier and/or more flexible for some of you.