I've been within the mobile and mobile marketing industry since 2009 (http://curtprins.com). I've seen more than my share of companies create apps that no one wanted and miss-use mobile tools at great expense to their budgets and customer loyalty. (Mobile apps can provide great value, but there's a wealth of other tools to explore first.) If you're not sure if or where to invest, a quick with me can ease your mind and likely save you a lot of heartaches and $$$.
Focus on apps/devices that make it easier to manage chronic diseases--asthma, COPD, diabetes, or other issues like infertility.
Managing these ailments can be cumbersome, and most patients fail at managing them. This is where mobile devices can offer the most benefit.
How long is a piece of string?
There are tons of different mobile messaging platforms available--some basic and others with steep learning curves. It all depends on what you need and what your budget can handle. Parse, Mixpanel and Urban Airship are the big ones, but there many niche players for gaming, retail, etc..
If you're a newbie, stick with a platform that offers an SDK instead of hard coding it yourself. This is the fastest way with the shortest learning curve, and many offer free trials and educational resources.
As for format, in-app and push messaging have their roles and one is not better than the other. For example, a weather app might issue a push every morning at 7am, and then offer in-app messaging for specific details and advertising offers. They each have their place. You should also not overlook email and text messaging, which are hugely effective at reviving lapsed users.
So it all depends on what you need. Stick with 3rd party services and experiment. Let me know if you have any further questions.