From your phone to your computer - automatically
Both Apple and Android phones will allow you to upload photos to the cloud automatically. In their free versions they don't offer a lot of space.
Apple iOS
The default is for your phone to store photos in iCloud which gives you up to 5GB free - this will likely not be enough space to store photos.
Google Android
Google gives 15GB but that also includes your email and attachments.
Microsoft
If you're a subscriber to Microsoft 365 (formerly Office 365) you get 1TB of online storage but you need to install OneDrive and tell your phone to upload the photos.
Your pictures in the cloud
All of the cloud services (and there are more than I mention) allow you to synchronise your online storage with your own devices. It can be just links (saving space but costing time) or the whole file (taking up more space but providing faster access and the actual file to hand).
Microsoft
When you install OneDrive it offers to backup Desktop, Pictures and Documents to the cloud automatically. It works pretty well and files deleted are saved online for 30 days (personal account) or 93 days (business account).
Apple
The default option is for your iPhone to save to iCloud and you must have an AppleId to use an iPhone so that it happens automatically. If you have a Mac the pictures will end up in the "Photos" app. If you have Windows they'll stay online unless you arrange to download them.
Again if you have an Android phone you will likely have a Google account and you can sync with Google Drive on your Windows or Mac computer.
Organising your storage
Year-month folders
iPhone allows you to organise photos into Year-Month folders and they will appear under Camera Roll on your computer. Android devices just do it by year.
I did find a quirky Windows little program to do that job and it would work for you moving from OneDrive\Pictures into your local Pictures folder. Using PhotoMove – Step By Step (mjbpix.com) You would want the pro version which costs $8.99 so well worth it.
Duplicates
dupeGuru is a tool to find duplicate files on your computer. It can scan either filenames or contents. The filename scan features a fuzzy matching algorithm that can find duplicate filenames even when they are not exactly the same. dupeGuru runs on Windows, Mac OS X and Linux.
Manual Organisation
There are possibly lots of tools for organising photos. I normally have mine in Year-Month order and that could work for your requirements of separating old and new - you can simply sort a folder by date, select the old ones (shift + click, top and bottom) and then drag them where you want. But it might be easier on a regular basis to use so and it would keep the folder structure
Naming and renaming batches of files
Photo files have names that convey very little information: img00123456.jpg is not that useful. So it can be very helpful to be able to select a group of files and change their name to organise them by location, holiday or by main subject. Or whatever else you want.
Here is a separate article on the subject: Renaming groups of files (butterworthconsulting.co.uk)