... Any recommendations on what HDD to buy for this model?
Also, on the iFixit video, the girl skipped the removing the casing from the HDD and inserting the new HDD, is that an easy job?
Finally, should I not increase the RAM first or is this not needed?
RAM is something that you can do at any time, and you don't need to do that at the same time that you would do other repairs. It's just a small access door on the bottom edge.
Hard drive can be your choice of any 3.5-inch drive - the capacity can be anything from 250GB > several TB

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What do I recommend? SSD. That would change how you replace the drive, as it would require changing how the drive is mounted internally. Again, that's something that you can find instructions fairly easily.
The video didn't skip anything, as far as I could tell.
There's a bracket, and a couple of pins that are transferred from the old drive to the new drive. It's just part of the replacement process, and the video mentioned both. A replacement hard drive doesn't come in a "casing", so I don't know what you were describing.
Or, you could find someone locally who can help you, or someone who works on computers can handle the task - particularly if you provide some information about the replacement through that iFixit site. I have a shop locally where I would be OK with the work, even through they rarely work on Macs, if at all.
But, that's up to you, too
