Storage should be fine to keep, though you will really want to add an SSD as well.
And yes, the GPU will go in the same way it does now, that's fine to keep for as long as you want.
The case may be worth looking at replacing. It depends what you'll ultimately think you'll want to have in your system. You'd likely want additional airflow, and i) the case's fan mounting options are limited and ii) if you were going to spend money to buy several fans for the case, that's already some money that can go to a whole new case with better/ additional fans already installed. And if you're rebuilding the core of the PC now, there's an argument for doing the case at the same time so you don't need to then move it all later when you get (say) a beefier GPU.
For your uses you'd probably be looking at an AMD CPU, e.g. R5 3600 or R7 3700x depending how much you want to spend.
AMD and their partners are about to release a new series of motherboards, B550. The current mainstream boards are B450, with X570 being high end. One of the key features of X570 motherboards is that they offer PCIe 4.0 SSDs. However, B550 will bring that to the mainstream segment, replacing B450. So if you don't mind waiting until
June you could go with one of those.
If you were keen to move forward now, something like:
PCPartPicker Part List
CPU: AMD Ryzen 7 3700X 3.6 GHz 8-Core Processor ($294.14 @ Amazon)
Motherboard: MSI B450 TOMAHAWK MAX ATX AM4 Motherboard ($114.99 @ Best Buy)
Memory: G.Skill Ripjaws V 32 GB (2 x 16 GB) DDR4-3600 Memory ($159.99 @ Amazon)
Storage: ADATA XPG SX8200 Pro 512 GB M.2-2280 NVME Solid State Drive ($78.98 @ Amazon)
Case: Phanteks Eclipse P400A Digital ATX Mid Tower Case ($98.98 @ Newegg)
Total: $747.08
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2020-04-25 07:21 EDT-0400
32gb RAM not for gaming, as 16gb is more than fine, but for your productivity uses. While the above isn't necessarily overkill (arguably you could spend more) you can still get a genuinely huge upgrade while considerably spending less.
PCPartPicker Part List
CPU: AMD Ryzen 5 3600 3.6 GHz 6-Core Processor ($174.15 @ Amazon)
Motherboard: MSI B450 TOMAHAWK MAX ATX AM4 Motherboard ($114.99 @ Best Buy)
Memory: G.Skill Ripjaws V 16 GB (2 x 8 GB) DDR4-3600 Memory ($79.99 @ Newegg)
Storage: Crucial P2 500 GB M.2-2280 NVME Solid State Drive ($64.99 @ B&H)
Case: Phanteks Eclipse P400A Digital ATX Mid Tower Case ($98.98 @ Newegg)
Total: $533.10
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2020-04-25 07:20 EDT-0400
With the 2nd option, it's worth noting that if you did want to massively increase CPU performance down the line, the board is expected to support the 16-core 4000 series CPU. It already supports the 16-core 3000 series. And the case would support closed loop liquid coolers appropriate for that kind of CPU.
Ofc the same is true of the first build, but the difference is you'd have spent less on the first CPU you'd have then replaced. That said, you may be quite unlikely to want to replace either CPU though especially the 3700x, given what you've been getting by on.
Case choice due to good airflow and a white frame, but ofc there are other options out there.