Can I build a 3 stage whole house water filter myself?
I've installed many filter systems this way, it's very common. 



(random from web; not my install)

Installing the filters themselves is pretty straight-forward, with a couple things to note:


Some people (including this picture above) install a bypass to the filters, but I'd actually not recommend that. It's too easy to accidentally open and have mostly unfiltered water coming into the house (water follows the path of least resistance). There's really no reason to have a bypass (you can't live without water for the 10 minutes it takes to replace the filters?).
If you have taps you want to get unfiltered water (eg, outside) then just split those off before the filters.
Installing a pressure gauge before and after the filter chain is a good idea to let you know when to change filters. You can install one between every filter if you want (and are willing to spend the money). 
Be sure to test screwing and unscrewing the filter before finalizing everything. In the above picture, for example, the water line below the filters will probably make it a bit of a pain to get the filter housing off. 
Install isolation valves before and after the chain. This prevents you from having to drain your entire pressure tank and house to change filters. 
If there is no pressure relief button on your filter (the red button on the top), install a sediment tap to relieve pressure (which you must do or you won't get the housing off). Not a bad idea to have one just for convenience anyway. 
Be aware each filter will reduce the flow and pressure. 




As far as safety of the water, that's a bit outside the bounds of this question, but just to highlight the point: it's a good idea to get your water professionally tested. 

Sediment filters are good, but they do not filter everything. Absolute 1 micron (as opposed to just "1 micron") filters are effective at removing some bacteria, but they are not as effective as a UV lamp. There are many other types of contaminates (iron, manganese, pesticides, etc) that are not treated by this type of filtration at all.