Your TOSHIBA MK3265GSXN has a thickness of 9.5mm. Most SSDs are 7mm, so practically any SSD will fit. You can go thinner, but if you go thicker it might not fit in the laptop's drive bay. (For instance, some large hard drives are 12mm thick.)
Here's
a selection from Newegg that will all work in your laptop. In your case, the only essential requirement is that you get the 2.5" form factor, not M.2 or mSATA.
Your laptop's spec is SATA-II while SSDs are SATA-III, but that just means you won't get the full speed out of the SSD that a newer laptop might. Don't worry about that, though, as it's still going to be noticeably faster than your current HDD.
As for brands, I typically recommend sticking to familiar memory chip manufacturers (Samsung, Intel, Crucial, Micron, Sandisk, Kingston, et al) rather than traditional disk drive manufacturers (Western Digital, Toshiba, Fujitsu, etc.). After all, SSDs are memory chips configured to look like disk drives, so I tend to trust manufacturers of memory chips to be right in their element here, more than manufacturers of magnetic media.
As for size, I wouldn't go smaller than 240GB, and SSDs larger than 512GB are still pretty pricey.
You mentioned, "I thought I'd upgrade the drive and get more space [than 320GB] as well." If the ~500GB range is beyond your budget, you might consider a ~250GB SSD internally and re-purposing your 320GB as an external data drive with an enclosure
like this. Still another alternative is to replace your DVD drive with the 320GB HDD by using a
second-drive media bay caddy, which allows you to have two drives in your laptop, but the downside is you'd lose regular use of your DVD drive.