Looking at the page you linked to, it looks like it will do the job for you. It has a 1/4" shank which might limit you down the road if you decide you like building things and start doing it all the time.
A rabbet bit goes against the work piece like this:
![picture of rabbet bit against workpiece](https://i.stack.imgur.com/4rjzT.png)
The dimension B is determined by the size of the bit; you can make it smaller by putting something between the workpiece and the bit; you could use another bit to cut out more, but that makes the whole job a lot trickier because you won't have the bearing on the bit to help guide the router. You have control of dimension A; you make the cut deeper by running the bit over the workpiece multiple times, cutting a little bit more each time.
Other items you'll need are safety glasses, a vacuum cleaner to collect sawdust (it's a carcinogen so don't want to be breathing it) and clamps to hold the work piece.
You need clamps because operating a router is a two-handed job: plunging the router with both hands means that you have even pressure on each side of the bit thus more control if it kicks and it keeps your hands out of the way of the fast-moving machinery.
If you don't already have something sturdy to clamp the workpiece to, you should probably think about getting a workbench. A dedicated routing table is another option, but probably more than you need right now (I'm just now getting to the point where I'm thinking I'd like to have one so can't advise you there).
Best Answer
If it is an interior door - the answer is no. Leave it as.
If it is an exterior door then yes for sure you should you should fill the cracks with insulation. For anything big enough - over 1/4" maybe - I stuff Roxul insulation in. Cram it in the gaps really good.
Then once you have the bigger gaps at least as filled as you can get with standard insulation you spray the smaller gaps and the outside parts of the larger gaps with spray foam insulation (cheap big box example is Great Stuff - not a recommendation just an example, however I do recommend Roxul above). Wait at least an hour - depends on temperatures - for spray foam to expand and slightly harden. Then scrape off excess foam sticking out. It is always good practice to get up close on each side and make sure you cannot see to the other side.
An added step that I have seen in cooler climates is to (flashing) tape the "cracks" after insulation. This for sure adds a wind barrier to the gap and might help with drafty doors/windows.