I apologise if this isn't all in simple terms: the problem is that some of this stuff cannot be explained without a certain level of complexity.
You should just be able to see the photo in your message app. If you can't that implies that either your message app is really bad (unlikely) or there is something wrong with your settings which is preventing the message from being downloaded.
So when you say "there was a photo attached somewhere", what exactly did you see in the message? If there's nothing you aren't comfortable sharing (e.g. names, phone numbers, message content) you could post a screenshot, but a more detailed description might help us understand exactly what's wrong.
But let's assume it's an MMS download problem. They download using your mobile data connection rather than WiFi. That should work even if you are connected to WiFi, but you could try turning the phone's WiFi off and then seeing whether you can download the message. If that doesn't work the second option is more complicated: mobile data connection settings are something called Access Point Names (APNs), and many networks use a separate APN for MMS as opposed to other data usage. For example I have 2 APNs, called "EE Internet" and "EE MMS", and if the second of those is not set up, or not set correctly, everything else will work except for MMS. Now setting up APNs should be automatic when you insert your SIM but it sometimes doesn't work, and with an old phone and a new account I would guess the risk of that is higher. So to check this you need to find your APN settings: they'll be somewhere in the mobile data settings, where exactly will depend on what android version you are running. Then find out what they should be for your service provider (via a web search) and check whether they are correct. If not you'll have to set them by hand, or maybe call your service provider and see whether they can do that for you (they may be able to do it remotely).
As for file managers or photo apps, you probably have an app called Gallery or something like that. That's an image viewing app, and should be set to show all images saved to the phone storage or SD card. However it won't show images that are only in the SMS/MMS database as that's in a different part of the storage that only message apps can see, so my guess is that a Gallery app probably won't help. Have a look anyway, but if it doesn't show anything you probably need to solve the download problem first and then select the photo in the message app and save it somewhere else before the Gallery app can view it (but once downloaded you should be able to see it in the message app anyway).
