What’s the procedure for putting a new SSD drive in my old Macbook Pro?

I would put the new SSD in a decent external enclosure, format it as Mac OS Extended (Journaled) then use Carbon Copy Cloner to clone your existing internal drive to it. Then test to make sure your MBP will boot from the external SSD. If all is well, then swap it into your MBP.

Carbon Copy Cloner v5 will run on El Capitan. I have relied on CCC for many years to make bootable clones and backup data. It has always worked perfectly.

You can download the El Capitan installer here -> How to download and install macOS - Apple Support but be advised that the macOS download servers have lately been unavailable.

A 2010 laptop can run up to macOS 10.13 High Sierra.

The simplest option may be to put the new SSD into an external enclosure. Download the macOS 10.13 installer and run it, but install it onto the new external SSD. After the first copy phase finishes, the installer will reboot the laptop which should boot to the new external SSD, but sometimes it will end up booting to the internal drive instead. If this happens, just restart the laptop and hold down the Option key immediately after hearing the startup boot chime. Select the external drive to boot from (should have an orange icon....not sure what the name will be at that time) so that phase 2 of the install process can complete.

After the install is completely finished, you will be greeted with Setup Assistant which will have you setup the new clean macOS 10.13 on the external SSD.....you will be given an option to Migrate from the old system....you should be able to select the old internal drive, or you could migrate from a Time Machine backup.

In fact I highly recommend you also create a bootable macOS 10.13 USB installer so you have more options in case something goes wrong. I would leave your original hard drive untouched because it is most likely worn out or even failing even if you don't notice it yet. Many times an OS install on an old hard drive will result in the drive finally being put over the edge. Plus leaving the original hard drive untouched will help you troubleshooting the system if you encounter any issues with the new SSD when it is installed internally.

Of course you could clone the original HD to the new SSD using CCC (great app which has never let me down for bootable clones). I do know that sometimes migration is better while other times a clone may be better....it can depend on the third party apps installed. I'm not sure whether there is any difference between migrating from the old system or from a Time Machine backup.

Create a bootable installer for macOS - Apple Support

How to download and install macOS - Apple Support

FYI, once you upgrade to macOS 10.13 you would get access to the online installers through Internet Recovery Mode (Command + Option + R), but it seems like Apple has quietly & secretly turned them off for reinstalling macOS so make sure to keep your original OSX 10.6 restoration DVD which originally shipped with the laptop and possibly keep a bootable macOS 10.13 USB installer handy as well.

You do have multiple options to choose from for this process. Sometimes it depends on your setup & apps, while other times it is just personal preference.

leroydouglas has good links related to replacing the drive in this post:

SSD internal hard drive - Apple Community

I tried cloning my drive when I installed an SSD in my 2008 MacBook - and I wished I had used Time Machine instead. I may not have done the cloning properly. I recommend putting the old drive in an enclosure (OWC Computing is a good source), so you can hunt down any files that didn't transfer. Then when that's done, you can reformat it and you have an inexpensive external drive.

The OS and other software are limited by the CPU, so you won't be able to upgrade any more than you could before the replacement, but it will run quicker.

I'd clone the drive using SuperDuper! then swap drives. Cloning replicates the drive 100%.

That assumes SD! will run on a 13 year old obsolete Mac.

If you upgrade to High Sierra, you may want to install the Firefox 115 ESR browser.

It's no longer on the main channel, and getting new features, but it's newer than the versions of Safari in El Capitan and High Sierra. Mozilla says they'll still be providing critical security updates until September 2024.