I have a Laptop with an SSD and an HDD. It came with Windows installed on SSD and since it is small, I added a HDD to store all my files and install all my games. I would like to install Ubuntu on a part of the HDD (without losing all the data on it. I have about 300GB free and can create an empty partition by "shrinking" the corresponding D: drive, so that should not be a problem) and want to keep using Windows as well as Ubuntu. I have seen several tutorials and questions about installing dual-booting Ubuntu and Windows, but all of them assumed they were installed on same disc. Is something different when it comes to installing Ubuntu on a different hard drive, will it still work and what are the possible problems and solutions?
You can absolutely do this. There are a few things you'll want to do first:
Next, you'll want to insert your bootable usb and reboot into BIOS. If you don't know how to do that, EasyUEFI is a nice tool to boot into BIOS and edit the UEFI order (the free trial is fine). To use that tool, install it and click on Power -> Reboot into UEFI.
Once in your BIOS/UEFI, select BOOT and move select the usb or move it to the top and save and exit.
You should then load a screen that has a few options to either try out ubuntu, install it directly, or a few other things. You can do either of the first ones as the live cd version of ubuntu will have a desktop icon to install ubuntu.
Once you start installing you will be asked a variety of settings questions (timezone, name, password, etc). Select whatever settings you want for those, although you probably do want the 3rd party software as that will make your life a bit easier.
Important: don't simply select to install ubuntu alongside the windows bootloader. Instead select the last option to select the custom drive (your HDD). Double click the empty partition you made earlier and set the mount point option to be root (/). Then click install and you should be good to go.
Here's a similar guide with screenshots if you get stuck on that last bit.