What are the options of upgrading Windows XP to a newer operating system?

You can't "upgrade" Windows XP to anything other than Vista, which would be a silly thing to do.

Depending on your computer's hardware, it might be possible for you to install a newer version of Windows.

Assuming that you were interested in Windows 7, your concerns are:

• Can you find a legal version of Windows 7 for sale (retail sales ended on October 31, 2013)
• Does your computer's hardware meet the minimum requirements for Windows 7 (1 GHz or faster processor, 1 GB RAM (32-bit) or 2 GB RAM (64-bit), 16 GB available hard disk space (32-bit) or 20 GB (64-bit), DirectX 9 graphics device with WDDM 1.0 or higher driver).  Preferably, your computer should exceed those "minimums."
• Are there Windows 7 drivers available for your computer's hardware

Interestingly, Windows 10 has the same minimum hardware requirements, but it's less likely there will be Windows 10 drivers for your old hardware.

To move from XP to a newer version of Windows you have to do a "clean install" (Microsoft calls this a "custom install").  Because this involves booting from the installation medium and formatting your hard drive, you first have to back up all of your data to external media and gather everything necessary to install all of your application programs (product keys and installation CD/DVD or downloaded installer file).

In the end, you'll be better off if you purchase a new computer with Windows 10 pre-installed.  You'll still have to transfer all of your data and install all of your applications (at least some of which will likely have to be upgraded because your old versions won't run on Windows 10).

P-W,

Your main concern is whether the investment in money, time, effort & blood pressure is worth it.  Any computer can fail and you might find that yours fails the day after you finish sorting it all out with a new OS - you might spend £100 and a week of your life but end up with nothing.

If you decide that your computer is going to last long enough for Windows 7 to be worthwhile then I suggest you also consider Windows 10.  If you believe [and this really is no more than an act of faith / blind optimism] that your computer is going to last at least another year, say, then there is likely to be no logical reason why you would not believe that it could last another three or four years.  A four year timescale extends beyond the Windows 7 end-of-support date so Windows 10 would then become your only option.

Windows 7 is currently being supported [security-related updates only] and will continue to be supported until 14 January 2020.  Windows 10 will be supported indefinitely. However, future Windows 7 / 10 updates might rely on assumptions about hardware capabilities that yours fails to satisfy so you might end up with a partially updated system with ever widening vulnerabilities caused by your old hardware.

Security is also of particular concern because older hardware & software might not be as resistant to intrusion as modern versions even with all possible updates installed.  Once your hardware & software is out of support by its manufacturer then hackers might start to attack its vulnerabilities and even the best anti-malware protection might fail to protect you.

You might also benefit from watching a new computer do some of your current tasks so you can see the improved performance you might get from buying a new computer instead of buying a new OS for your old one.  Just for example,

• Modern WiFi & Ethernet cards operate at far higher speeds than were possible when XP was still current - you are likely to be astounded by the difference in speed.
• Modern computers are also likely to astound you with their processing speed.  I do not know what you use your computer for but I do some minor video editing and what takes my old computer a few minutes is done in a few seconds by a modern computer.  This time comparison would also apply to other processor-intensive tasks.

Your best bet is to search your computer maker's support site & user forums for reports by users who have installed Windows 7 / 10 on that computer model.  In addition, search this forum for your computer model in case anyone has reported useful information about installing Windows 7 / 10 on it.  You could also search the internet generally but that would take a long time and might get you nowhere.

You should check the Windows 7 system requirements and run the Windows 7 Upgrade Advisor.  There is no upgrade assistant for Windows 10 that will run on XP but the Windows 8.1 system requirements are practically the same as the system requirements for Windows 10.  Therefore the Windows 8 Upgrade Assistantresults will also give you a good idea of the system’s compatibility with Windows 10.  Early versions of the Windows 10 system requirements also included a need for processor extensions SSE2, NX, PAE; I have no idea if these are still relevant or not. 

If your software is also old then search the publisher's support site & any user forums for reports from users who have used your version with Windows 7 / 10.  Upgrading / replacing your software also adds to the money, time & effort you need to invest.

Whilst some old computers can work with new OSs, you should not expect everything to be perfect.  I have installed Windows 10 on an 11 year old Dell Inspiron 9300 and the only shortcomings are that the PowerCfg utility -BatteryReport command does not work and Task scheduler cannot wake the computer from sleep or from hibernation [even with the correct settings made].  I can live with this but you might find shortcomings [after all your money, time & effort] that render the computer useless for your purposes.

Whatever other guidance you find, you will need to search for hardware drivers - see Where to get hardware drivers.

 I succeeded in this endeavour but it took a fair amount of experimenting & backtracking.

If you decide to go ahead then make sure you are able to get back to square one as a Windows XP computer might remain useful as a backup computer even though you cannot connect it to the internet or any other network / other computer.  You can find software to create "system images" and they would be your best method of reversion.  Do not proceed with anything until you have made a system image and learnt how to restore it.  I did not know what system images were until I started using Windows 7 - they are a complete copy of your hard drive that can be restored to it to re-establish the computer in exactly the same state it was in at the moment the image was made.  Modern OS include the ability to create system images but XP does not have this.  Software for making system images does exist that can run on XP - I use a free** version of Acronis True Image on mine, Macrium Reflect [free] is another imaging utility, Paragon Backup & Restore [free] is another and there are lots more.  In general, you will need to make both a system image & the boot disk / rescue disk required by your imaging utility.  [** Free versions of Acronis TI are available for computers with Samsung-Seagate & WD drives inside or just connected to them].