As Palcouk says, there is no discounted upgrade (if you are looking for one).
The Office applications are setup to allow easy upgrade. Simply uninstall Office 2003 (using Control Panel. > Programs and features), then install Office 365 or 2019. The new install will pick up the configuration information from the old install.
If you need them, here are links to more specific instructions.
Uninstall Office 2016 / 365 – 2007
https://support.office.com/en-us/article/uninstall-office-from-a-pc-9dd49b83-264a-477a-8fcc-2fdf5dbf61d8?ui=en-US&rs=en-US&ad=US
Method 1: Uninstall from Control Panel (or Settings dialog)
Method 2: Completely uninstall Office with the uninstall support tool (SetupProd_OffScrub.exe)
Method 3: Remove manually (Registry Hack)
The Method 1, “Normal uninstall”, is designed with the assumption that you will be re-installing Office. It does NOT delete everything. It leaves configuration files like NORMAL.DOTM and other templates etc. It also leaves configuration
Registry entries. Normally these “carry-overs” are safe. But if method 1 didn’t fix your problem, they may be the cause.
Note:Method 1 includes links to Office version specific re-install instructions
You now have 2 licensing options for Office. The "new & improved" license is Office 365. It is a keep-paying-forever "subscription. You can pay for monthly (more expensive) or yearly subscriptions. They add some "bright and shiny" trinkets to make the extra
cost more paletable. The other license option is the one you are used to, one-time payments. You pay once and have use of the license "forever". MS also (dis)"improved" this type of license. You can now only install on one computer at a time (previously
it was 2 or 3 computers, depending on the license). And now the license is only good for 7 years, down from 10.
I Need Office, What Should I Get: Pros vs Cons – WIKI – buy Office - “Free” Office
http://answers.microsoft.com/en-us/msoffice/wiki/msoffice_install-mso_other/i-need-office-what-should-i-get-pros-vs-cons/2e75ea21-f8d0-4cde-9764-13b02cde46a7?tm=1478718433145
Check out the wiki I go into more detail outlining these options:
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Stand alone Office Programs
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Office 2016 Home and Student
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Office 365 Home / Personal / University
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Office 2016 Home and Business
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Business Office 365, several possible bundles
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“Office in the Windows Store” aka Office Hub
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MS Office Online
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Non-profit license
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WordPad
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Works 9.0
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Free/low cost Office Alternatives
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Where can I buy Office product keys
You will have to decide for yourself. You can to trade off long term cost and features If your needs are very simple, you won’t need to spend much or work with one of the free options. But if you are a more “advanced” user, using multiple
Office applications and sharing documents/files with others you will have to pay for one of the Office 365/2016 bundles.