I have 2 drives on Ubuntu. Can I use them together simultaneously?

I do apologize. I am new to this IT customization.

 sda      8:0    0 465.8G  0 disk 
├─sda1   8:1    0  19.5G  0 part /media/
├─sda2   8:2    0  13.7G  0 part /media/
├─sda3   8:3    0   4.1G  0 part 
├─sda4   8:4    0   428G  0 part /media/
└─sda5   8:5    0   500M  0 part /media/
sdb      8:16   0 119.2G  0 disk 
├─sdb1   8:17   0   300M  0 part /boot/efi
└─sdb2   8:18   0 118.9G  0 part /

Results of cat /etc/fstab :

ubuntufun@shisui:~$ cat /etc/fstab
# /etc/fstab: static file system information.
#
# Use 'blkid' to print the universally unique identifier for a device; this may
# be used with UUID= as a more robust way to name devices that works even if
# disks are added and removed. See fstab(5).
#
# <file system>             <mount point>  <type>  <options>  <dump>  <pass>
UUID=9A47-5A41                            /boot/efi      vfat    umask=0077 0 2
UUID=b41f7887-d80b-4afc-be14-d1955a56b44a /              ext4    defaults,discard 0 1
tmpfs                                     /tmp           tmpfs   defaults,noatime,mode=1777 0 0

I setup Ubuntu on my desktop which contains 2 drives in it. I do believe I partitioned them (accordingly with sdb2 120GB SSD having the home folder, sda4 with 428GB currently free) when I installed Ubuntu and totally removed Windows. My intention was to use the SSD for running programs at a lightning pace and the slower hard drive with more space would contain folders that would be ideal for storing certain files/folders.

Now that the SSD is full to capacity, I cannot copy and paste from the SSD to the other drive (I am still new to Linux and have been slowly learning about its setup). I can see lib files, however neither copy and paste or creating a new folder options are not available on the larger drive.

Yes, you can. You have to be administrator, i.e., have sudo permissions, in order to manage storage media. Drive management and ownership requires some reading up. This cannot be covered in an answer here. so what follows are just some leads to get you started.

On Ubuntu, partitions are not automatically mounted after startup, unless they specifically have been configured to do so. On a default install, only the partition containing the system, i.e., the root partition, is mounted. It is mounted on the directory '/', the upper topmost folder in a linux system.

On Ubuntu, other partitions can be mounted before use by clicking the icon representing the partition in the left pane of the file manager Files. These partitions are then mounted by the system under a folder /media/<userlogin>/<label>.

It can be desired to also have data partition automatically mounted during startup. Power users can edit the system configuration file /etc/fstab directly, but you can also use the utility "Disks" to configure a partition to automatically mount. In that tool, select the partition in the map, then click the cog wheel and select "Edit mount options". You must specify a mount point, i.e., a folder in which the entire partition will appear after mounting.

Next to the aspect of connecting a partition to your file system (mounting), there is also the issue of ownerships and permissions. By default, a user can only read and write in folders under his home folder (typical /home/<login-name>). Before users can write anywhere else, either they must be made owner of the folder, or they must belong to a group that has write permissions for the folder. Alternatively, the permissions for others must be fully opened, but that is not safe practice.

A 120 SSD is plenty, both for the operating system and for storing user data that you use daily. In your case, I would work mostly from the SSD, and use the HD for storing archive data or large files like pictures and music. An elaborate scenario of a user arranging data in different partitions, and using links to make them accessible directly in the home folder is available here.

I suppose you want to use /dev/sda as a 'data partition', as windows users usually put it.

It's quite straightforward to achieve. The way you think about it, however, has to change a little.

It's like you have a bunch of bags, and want to put stuff in them. The linux way is first to sew the bags up to your jacket, as if they're genuine pockets.

So here we are talking about mounting partitions to where you store stuff which most often refer to '/home/...', or dirs in your HOME. To complete the anology, '/home/...' denotes the sewn pockets, while '/' denotes your whole jacket, or the ROOT PARTITION (which in your case is /dev/sdb2).

For example, you can mount /dev/sda4 to /home/data as one pocket, and mount /dev/sda{1,2,3} to /home/{video,music,doc} as others.

Now the task is how you mount spare partitions to /home/.... To achieve this, you need to:

1. Mount /dev/sda{4,1,2,3} to somewhere like /media/sda{4,1,2,3}, repectively.
2. Move current data in /home/... to /media/sda{4,1,2,3}, respectively.
3. Write all the mount assignments to /etc/fstab, one line for each partition, similiar to the original lines. (You need to learn how mount works, including its command syntax.)

But before you begin, it's most important to think over and understand what you are about to do.