You’ve just purchased your first bagpipe and you’re wondering how to care for it.  Of course, there are many differing opinions.  Here are some basic thoughts. 
There are two approaches; one being pro-active and the other being reactive.  A pro-active approach is best however may not be possible until experience is gained.
First and foremost, try to keep your bagpipe in a more-or-less consistent environment.  This means not too hot and not too cool.  This also means not too wet and not too dry.  Especially in North America we can experience wide swings in temperature and in relative humidity.  This makes things very challenging.
With changes in temperature and with moisture, your drones are expanding and contracting in size.  The natural make-up of the wood will generally allow this to take place without causing issues or problems.  Natural oils help to keep the wood pliable and therefore less susceptible to cracking.  Too much oil and you’ll end up with a dull sounding bagpipe.  You can research all this and be as technical as you wish.  In my experience, I generally give my pipes an exterior oiling a couple time each year.  Just as importantly, I blow my bagpipes daily (or every other day) just to make sure they’re well.  It is much easier to maintain a pipe properly than to bring one back from a place of neglect or abuse.
Your reeds are a great indicator of how things are keeping.  I generally inspect my chanter reed before and after playing.  If the reed is holding too much water at the end of playing, I might hold a tissue to it to absorb surface moisture.  I might also store it in a manner that will allow the moisture to dissipate gradually.  If you dry it too quickly you run the risk of changing the shape of the blades and damaging the characteristics (volume, pitch, and strength) of the reed.  If the reed is a bit dry, I might introduce a moisture to it before storing it, either back in the bagpipe or within a reed protector. 
Drone reeds are also an indicator of what’s going on inside the bag.  You don’t want water lying on the surface of reeds.  It will just trap dirt and eventually impact sound and performance.  After playing I remove the drone reeds and examine the bores of the drones.  If moisture is visible on the inside bores I use a soft pull-through to remove it. 
Most pipe bags will now allow entry through a zipper.  This allows for instant assessment and adjustment of conditions impacting your bagpipe.  If you’re finding the pitch of your reeds heading skyward, it’s probably time to introduce a bit of moisture to the bag.  This isn’t often the case.  You’re more likely to find too much moisture in the bag, which translates to too much moisture to your reeds and too much moisture to your drones.  There are many things you can do to counter this situation.  Again a bit of research and through trial and error you’ll figure out what works best for you in your particular environment and piping conditions.
The very best thing you can to is to pay attention.  Watch and listen to other pipers.  Listen to your own bagpipe and pay attention to any changes in sound and behavior.  Figure things out.  The GHB is complicated however it's also completely logical.  For every issue you experience there is a solution.
I hope this has been somewhat helpful.  It's perhaps impossible to cover every possible scenario.  If you pay attention to the big issues, you'll gradually learn how to identify and deal with all the subtle and obscure idiosyncrasies of this amazing instrument.
Care of your Bagpipe