I've been playing Skyrim via Steam Purchase, but I want to buy the whole Elder Scrolls Anthology for not only the Skyrim DLCs but for all the other games and their DLCs. Will the saves I have on the Steam version be there when I play Skyrim from the disk or will I have to start over?
All content for Steam is saved either on your hard drive or in the Steam Cloud. Either way, your content will be available after the DLCs are bought.
If the Anthology is offered Via steam then there will be no issue. As far as I know, Skyrim is steam only.
In the case that it isnt, you can always navigate to your savefile and copy it
Yes, you will keep your save. You just won't need the disk any more, so you can throw that away (not literally) and when it is downloaded you then just go to your games and start playing and start having fun with your game.
A general rule with all PC games -whether steam is involved or not- is that it will save to your Hard Disk (or a server if it's an always-online type)
If it's on your hard disk, there is normally a way to find your save file so you can copy it to a different location if another copy of the game would look in a different place. I have used this tactic for backing up saves as well as using the same save file on different physical PCs.
Steam has the steam cloud that supported games have their save files copied to so if you would change computers you keep your saves automatically.
In your case it's likely that if you buy the anthology through steam (or activate it on steam) it'll literally class it as Skyrim + DLC so you'll see no difference in your library or it'll overwrite your skyrim entry with the entry for the anthology. So your save files will likely not be affected in the slightest.
You will probably need to copy your Skyrim Steam save files into the folder that your disk install of Skyrim uses to store its save files (a google search will find both of these directories easily). It may be that they both use the same directory by default (probably My Documents). If this is the case then you should only need to worry about the point below.
You should also ensure that your steam and disk version of the game are at the same version level (patches), not having these level can cause havoc with save files and can corrupt them.