I am a very new Debian 10 user, just started a few days ago. During that time, the sudo command didn't work but I got it fixed. However, when I went to the Debian Softare Store, if gave me an error that said, "Unable To Download Updates, failed to refresh cache" I didn't do anything to fix this, as I didn't want to mess anything up.

Never heard of the Debian software store. Have you opened a terminal and typed and if so, what happens?

it says this

Reading package lists... Done
E: Could not open lock file /var/lib/apt/lists/lock - open (13: Permission denied)
E: Unable to lock directory /var/lib/apt/lists/
W: Problem unlinking the file /var/cache/apt/pkgcache.bin - RemoveCaches (13: Permission denied)
W: Problem unlinking the file /var/cache/apt/srcpkgcache.bin - RemoveCaches (13: Permission denied)

Sorry, my fault: you will need to run: and provide your user's password when asked.

its ok

also it say this

Hit:1 https://dl.winehq.org/wine-builds/debian buster InRelease
Get:2 https://packages.debian.org/buster/synaptic buster InRelease
Err:2 https://packages.debian.org/buster/synaptic buster InRelease
Clearsigned file isn't valid, got 'NOSPLIT' (does the network require authentication?)
Reading package lists... Done
E: Failed to fetch https://packages.debian.org/buster/s...ster/InRelease Clearsigned file isn't valid, got 'NOSPLIT' (does the network require authentication?)
E: The repository 'https://packages.debian.org/buster/synaptic buster InRelease' is not signed.
N: Updating from such a repository can't be done securely, and is therefore disabled by default.
N: See apt-secure(8) manpage for repository creation and user configuration details.

It appears you have added repositories to your . Is this correct? None of the ones listed above are I believe, default.

yes its correct

Remove or comment out the entries you added and then update. I don't use Debian so can't help solve why those entries do not work but I am sure there is a solution.