If you are using Lightroom Classic, that means that you already have the tools available to you in your subscription, i.e. Lightroom Mobile on your phone, and cloud syncing. The basic process is that after installing LrMobile on your phone (login using the same subscription account details that you use on your Classic system), you can set things up so that any new pictures you take are automatically added to the LrMobile app, and from there they automatically upload to your Adobe cloud space. In LrClassic you would need to turn on cloud syncing, and thereafter a copy of all images added to your cloud account are automatically downloaded and added to your Classic catalog. A few things to note:
1. Before turning on sync in LrClassic, open the Preferences>Lightroom Sync tab. There you'll see an option to change the location for Classic to store those downloaded copies from the defaul (which is an obscure location on your system drive) to one of your own choosing. You can also choose to have Classic store them into one of the standard Lightroom date-based folder schemes.
2. On the phone you can continue to use the native phone camera, or you can use the camera app within the LrMobile app (or you can use both). All pictures taken with the LrMobile camera are automatically added to the app and synced to the cloud, and there's an app setting which allows any new photos taken with the native camera to also be automatically added to LrMobile. Existing photos on the phone can be manually imported to the LrMobile app.
3. Assuming you have the standard Photography Plan subscriptions, you'll only have 20GB of cloud space, so you'll likely need to develop a workflow that allows you to stay within that 20GB limit (or pay to upgrade to 1TB). Many users happily operate within the 20GB allowance, and the trick is to wait for the new photos to sync down into Classic and then to "unsync" them. That deletes the photos from the cloud, but retains them in Classic (you can "unsync" them in Classic simply by removing them from the All Synced Photographs special collection which appears in the Catalog Panel in Classic when you enable syncing). One unsynced, and thus deleted from the cloud, you can then sync them again but this time from Classic (add them back to that special collection or add them to a synced collection), whereupon Classic uploads a smart preview of the image to the cloud. From the cloud, the smart previews are then available on your phone and any other mobile devices using LrMobile (there's also a desktop app and a Web browser app that could also be used). Although only 2560px on the long edge, they are generally suitable for viewing and light editing, and the good thing is that smart previews do not count against that 20GB allowance....so you can sync your entire Classic library to the cloud without taking up any of that 20GB allowance. Once in the cloud, they of course are then available on your phone etc.