My girlfriend calls me the Google Photos peddler since I recommend it to everyone. It's a wonderful service and so versatile. I reached my storage limit since I decided to upload photos with the Original Quality option. I bought the cheapest plan, 100gb a year for 20 bucks. Pretty good deal and as someone who dabbles in photography and occasionally make poster prints, I do prefer having my photos backed up uncompressed.
The general consensus is that Google Photos does a fantastic job at compressing while keeping photos looking good. And hey free unlimited storage is awesome. But since I want to keep my pictures pristine and untouched it's worth the storage space. I'm not sure how much you're paying for Flickr but I'm sure it's similar depending on how much storage you need.
As for the migration, it won't be the easiest thing in the world. There might be some cloud transfer services like this, http://www.cloudsfer.com/features/.
Or you could take those photos and put them in Google Drive so they slowly pour over to Google Photos.
You have a few options...
There are a few 3rd party apps that seem to be ok (I've used Cloudsfer which worked well http://www.cloudsfer.com). For what it's worth, it may be cheaper just paying for a supported product
Learn the API, if you're familiar with programming you could whip something up in a variety of languages. Documentation can be found here - https://developers.google.com/drive/
There is another way that works, although a little manual, where you upload all the user folders (using the Google Drive app) to one account then share their respective folder to each user, you can then either train the user to add the shared folder to their Drive or do it for them. The process would look something like this...
Create or use an existing account that will be used to hold all the user data folders
Install the Google Drive application on a machine with enough storage to hold a copy of the user data folders
Copy all the user data folders to the Google Drive folder on machine - this will start the sync up to Google
Once the sync has finished, login to the Drive web interface and share each folder with the respective user
Once you have shared the folder, you or the user will need to log in to their Drive account, navigate to Shared with me and click Add to my Drive" on the shared folder. This will add the folder to the users Drive.
I have been using Mirrorace to push opendir LinuxISOs to a GDrive for the past 4-5 months and it has been great. See, here.
edit: there is also Cloudsfer. The speed is decent, but interface is ugly and non-informative, and free account limit is 5GBs. There is no way to delete an account after finishing that, unless you email them for it. If you can't delete account, it says your GDrive is added to a Cloudsfer account already. So, if you used a disposable email, no double dipping on free account option.
Dropbox will backup all pics, download, camera, etc. Though it will not upload MMS pics unless saved on device.
Cloudsfer will transfer files between cloud services.
Edit: Content
actually, the way I migrated from dropbox to onedrive was to use http://www.cloudsfer.com/ and it went pretty well. it happened fairly quickly.
Not sure when you used it last, but my recent usage has been fairly smooth. I'm not a big power user though...