Helo,
To start with, an OS upgrade is very unlikely to make your computer run faster. If you truly wish to upgrade then we don't know enough about your machine to be able to tell you exactly which systems you can use. The various ways of checking for software may only show updates to the current main system version you are running (e.g., 10.4.8 --> 10.4.11), not upgrades to a higher system version such as a jump from El Capitan (10.11) to Catalina (10.15). Note the difference between an update and an upgrade.
We can provide more specific advice if we know more about your computer model, year, and specifications. To get more information about your computer choose Apple menu > 'About This Mac' in the upper left corner of any window, then "More Info..." or "System Report". Copy and paste the information here, but omit the serial number and UUIDs (if present). Please also include the year which may not be provided in this information. More about this in "About System Information [Profiler] on your Mac" - https://support.apple.com/HT203001
You can also use the computer information and the links below to yourself determine what is possible with your computer.
Click on this link to read how to get different versions of macOS - https://support.apple.com/HT211683
- This article also has links to system requirements for each OS version which you should check against your exact computer model. Note some of the model identifier articles are outdated. For any model produced in the last 5 years or so you should use the system requirements links for each OS version instead. This web site also has a reliable presentation of model and OS version compatibility: https://eshop.macsales.com/guides/Mac_OS_X_Compatibility
- Make sure you have more RAM than what is noted as "minimum" for any particular system or your computer may run slowly.
- You may find you need to use Safari to get links for High Sierra and later using the Mac App Store application to open a functioning page.
- Sierra, El Capitan, and Yosemite download as a .pkg. Click on this to have a converted version placed in your Applications folder as an "Install OS..." application. Run the installer and follow the directions.
- Upgrades may take some time (up to an hour or more) to install, with restarts and blank screens.
Back up your data in case something goes wrong --> Back up your Mac with Time Machine - https://support.apple.com/HT201250 or use a cloning tool.
Check if your old apps will still work with with the OS you want to install - https://roaringapps.com/apps This is particularly critical if you are upgrading to Catalina or newer, as older 32 bit apps will no longer work.
The other way to look at this is to see what is making your ocmputer run slowly. It may simply be everything else has sped up in the world off technology and putting more demands on your old computer. There's little that can be done about that. Not nothing, just little. There could, however, be something specific causing this so I suggest you run this tool and post the results here. I have provided a link to a discussion telling you how to do this.
Run this analyzer and post the results here:
Etrecheck - http://www.etrecheck.com/
EtreCheck on the Mac App Store - https://itunes.apple.com/app/etrecheck/id1423715984?mt=12
See http://www.etrecheck.com/faq for information about the free and power user versions.
Using Etrecheck - https://discussions.apple.com/docs/DOC-6174
Extra tips on getting the free trial version of Etrecheck, settings to have it check all parts of your computer, and how to post long reports to the Apple Support Communities window: https://discussions.apple.com/thread/250599211?answerId=251149996022#251149996022