Welcome to Android Central! One easy way to move your photos would be to back them all up to your Google Photos in the cloud:
https://support.google.com/photos/answer/6193313?co=GENIE.Platform=Android&hl=en. Once you back them up, you can access them over the web using Google Photos on your new phone, assuming you log in with the same Google account. This also keeps your photos safe, because if you only have them saved to your phone, you run the risk of losing them all if the phone dies. If you choose the "High Quality" setting (as opposed to the "Original Quality" setting), you get unlimited photo storage in the cloud.
If you really just want them locally stored, then at least back them up to your computer hard drive. Connect the old phone to your computer, swipe down the notification panel, tap the USB icon, and make sure MTP (or File Transfer) is selected. Then, on the computer, open Windows Explorer, select the phone in the device list, and navigate to the /DCIM/Camera directory. Select all of the photos, then drag and drop them to your computer hard drive. You can then copy these photos to the new phone by connecting it to the computer and reversing the procedure.
Games can all be reinstalled to the new phone--as long as you use the same Google account, you don't have to re-purchase games on Google Play. However, game progress might be difficult to transfer.