Cloning
You can clone the partition with Clonezilla or dd. I would recommend Clonezilla.
Copying with rsync
An alternative (to cloning) is to create a partition with the ext4 file system: copy the content of the earlier root partition to this partition with rsync, when booted from another drive.
sudo rsync -Hav old-root-partition-mountpoint/ new-root-partition-mountpoint
Additional tasks
But cloning/copying the partition is not enough. You must also
create a swap partition or swap file that matches the corresponding line in /etc/fstab
install a bootloader (which will be different depending on the boot mode, UEFI or BIOS (alias CSM or legacy) mode.
If you have some experience of these things or are prepared for some trial and error, good luck :-)
Fresh installation
Otherwise I think, it will be easier to make a fresh installation and later on copy you personal files (documents, pictures, multimedia files) from the old system to the new system.
If you wish, you can create a separate home partition with the ext4 file system: copy the content of the earlier /home directory to this partition, and during the installation select 'Something else' at the partitioning window and let the installer use this partition for /home alongside the root partition (/) and swap partition.
This will preserve the personal files and tweaks (those, that are stored in the home directory (or directories)).
I doubt you will need 300GB for software. Almost certainly most of that will be needed for data. The simplest way to get more space on your system is to add a Linux partition (ext4 is a good format to use) to the HDD. Looks like you have about 200GB unallocated. You can use Gparted to make and format that partition. Then you add a new mount point somewhere like /data and add an entry to /etc/fstab to mount the new partition at boot time.
Alternatively you may be able to shrink your windows partition on the SSD and expand the Ubuntu partition. This is not too difficult to do but you may not be able to give up enough space from Windows.
There are plenty of detailed answers on Ask Ubuntu and on the wider internet about how to do these things but if you get stuck ask another question. Good luck with your course!
Best Answer
The way to solve this problem really depends on how much you care about your windows partition.
Option 1 - you don't really care about your windows partition:
Shut down your PC with the DVD drive open
Put the Ubuntu live DVD in and boot from the DVD
When your trial Ubuntu boots up start the program called "gparted"
Use gparted to increase your Ubuntu partition. The software is pretty easy to figure out. Right click on the partition of interest and select "resize/move". Make sure you are cognizant of where the partition has data (data is yellow and "assumed" empty is white) and avoid shrinking any partition where there is no white space left!
If at this point your Windows becomes broken, you can always use your Windows install cd/dvd to repair the Windows O/S (on your Windows partition).
NOTE: you may run Gparted from your Ubuntu partition, but then it may give you a hard time because you are trying to do work on a live partition (you have to unmount, the drive you are using, etc...). So you might get away without using a live CD, but why mess around when a live CD is the better way to go? (opinion only)
Option 2 - you really care about your Windows partition:
I don't use Windows 10, but in Windows 7 there is a way to shrink the size of your partition. Right-click "my computer", then select "manage" and from there you go to the "Storage" and open "Disk Management". There you will want to reduce the size of your windows drive. this is important to make sure you create empty HDD space for your Ubuntu to grow onto.
follow the steps outlined in the section above.
NOTE: The Windows work you do may create space on the other side of the hard disk. In that case you will need to use gparted to move the partition, or grow it in the other direction and then shrink it back to the original size.
NOTE: before you do anything back up your data!!! You will grow your Ubuntu partition and that will be easy. If your windows breaks, use the Windows install DVD to fix it - easy. BUT if you overwrite your data that is held by Windows... you will never get it back.
More info can't hurt. Here's another guy with a very similar issue: Give more Hard disk space to Ubuntu