First off, if the keywords are pretty related, having similar synonyms, then it can be pretty easy to rank for all three.
For Example, I used my home page to target: Car Dealerships City, City Car Dealerships, City Used Cars, City State Car Dealerships, etc.
Since all of these are so relevant to each other, it made things as easy as just talking about my business in my city-state.
However, my internal pages are very specific: City state auto repair, city state car rentals, etc.
Point is, if your top three keywords are about the same topic, then it can be easy to rank for all three on the home page. However, if they are different, and highly competitive (Diets and Gyms), then you are going to need to split up the content into different pages to rank well.
As for which pages you should focus on, it depends on your current rankings, and traffic. If you currently rank pretty well overall for your home page, then get it bumped up to the top to increase traffic. After that, start working on all the other pages.
If you aren't ranked well yet, and you don't have much traffic, then go after a niche phrase with decent search with all your internal pages. This will help your bottom line more - easier conversions - and still help your overall domain authority. I'm having to turn around, and do this with all of my websites that I took over (In-House SEO).