In my experience it's not as simple as "Just download the app".
The problem is that, like all public transport systems, it's (a) set up for the locals and (b) infused with historical stuff. In the case of Paris, there's also the turf wars between SNCF (the national rail company, which runs the RER suburban rail lines), RATP (the Paris transport company, which runs the buses and the metro), and a consortium called Île de France Mobilités, which is needed so that everything doesn't grind to a halt when you get to the formal boundaries of the city of Paris. It's at least as complex as TfL in London.
In particular, it helps to be aware that a lot of the branding revolves around the word "Navigo", which was originally a smart card available only to residents on which you load credit and use it to travel around, like Oyster circa 2010. That branding has now leaked into various other parts of the system, which can cause confusion because you don't have a physical Navigo card, you don't need one, and you aren't going to get one, but you are still going to be buying "Navigo tickets".
What worked for me (on Android, but I assume the same apps are available for iPhone) was:
- Download the app called "Île de France Mobilités". Create an account. Connect that account to the app. This will involve quite a bit of faff (accepting all kind of permissions, getting codes from your e-mail, etc).
- When you try to buy "tickets on my phone", you will eventually get to the point where it tells you need another app, called "Mes Tickets Navigo". This is because as well as "Île de France Mobilités" there are other apps that also let you buy Navigo tickets (because of the above-mentioned turf wars). So you need to install that. Note that it does not like things like power-saving mode or being put to sleep in the background.
- Now you can go back to "Île de France Mobilités" and try to buy some tickets. I suggest you start with a single €2.50 Métro/bus ticket.
Basically, "My Navigo Tickets" runs in the background and is the app that talks to the ticket barriers over the NFC (wireless) connection of your phone. "Île de France Mobilités" is the app that you will actually use to plan routes, buy the tickets, etc.
I strongly recommend that you do all of the above steps now, and then again on the day before you travel (obviously without actually re-installing the apps, but do go through the apps to the purchase part and maybe buy another single ticket, just to make sure that the logins work and the cookies are not expired, etc).
When I used this last autumn I found that checking in and out of the ticket barriers always worked OK (the "Mes Tickets Navigo" app is pretty robust and works when there is no data signal, which is still not ubiquitous on the Métro), but "Île de France Mobilités" was sometimes a bit flaky. There were moments when I could buy an airport ticket but not a single, and then 10 minutes later it would work. So buying tickets is probably best done when you're sitting comfortably at a café, and not when you have none and it's raining.
As far as I know the tickets cannot be added to Apple Wallet. I certainly didn't see a way to add them to Google Wallet. That's what the "Mes Tickets Navigo" app is for.