Yes, the SATA standards are compatible to each other, however the slowest involved connection type will always be used. This means that connecting a SATA 6Gb/s SSD to a SATA 3Gb/s controller will severely limit its peak performance.
More information can be found in the FAQ “What is the difference between P-ATA (IDE), S-ATA, S-ATA II and S-ATA III?”
You can use the disk management utility built into Windows 10.
You have to pay attention, that your hard drive uses the same connection type and form factor or height.
With the very common 2.5” hard drives you have to check whether your notebook still uses the old PATA standard or SATA and also you have to check what height is compatible since there are multiple heights. 7mm, 9mm and 15mm are the most common ones.
For SSDs there are also small form actors like mSATA or M.2. With the latter one you have to check if the slot supports the SATA and/or PCIe (NVMe) protocol.
The only thing that is not limited is the capacity and speed of your new drive.
The difference is in the type of connection and its speed.