I'm looking to buy my first house with my fiancee this year, but we
don't have a clear idea of the process or all the costs that are
involved. I've tried looking online to sort it out, but there are
thousands of pages on the subject and its all very overwhelming
(especially combined with our stressful jobs). We've tried to talk to
our parents about it, but its been so long since they bought their
houses they don't really remember all the details.

What I'd like is to find some sort of organization that would actually
have human beings I could talk to for some good advice, so I can ask
questions when I'm confused. It seems to me that some service like
this must exist, but I have no idea where to start looking. Then
again, maybe I should just save my money for the house and put the
extra effort in to decipher all the info myself?

Go talk to a real estate agent.

There really are no services that do this since so many of your
questions will be follow up or tied to specific local information that
it would be hard for a person to step in and answer them

As a first time home owner who is closing in three weeks, My advice would be:

1A) Talk to your friends and get references for realtors  - Realize
that different people will work well with different realtors.

1B)Start looking at neighborhoods to determine where you want to live
- take a look at Realtor.com to determine approximate pricing.
 
1C)Make a checklist of things that you are looking for ie:
   Number of bedrooms,
   Number of bathrooms,
   Square footage
   Appliances (dishwasher, central air vs wall units, etc)
   Lot Size
   House type (basement etc).
 
Pull up a mortgage calculator and start trying to figure out how much
you can afford monthly -remember in addition to the house price, there
is insurance, Additional bills like water/trash that you might not
have had before- the big one is property tax.

You can do all of these at the same time- 

Next call the realtor and talk to them about finding a place you
should be able to talk to them without a commitement.

From that point, your realtor is your advocate who should help you
through the process...

Some useful links

http://www.realtor.com - Good site to look up listings
http://homebuying.about.com/ - Good overview of the process.

While the two previous comments are useful I would also add that a discussion
with a good real estate lawyer is almost essential for a first time buyer.

You will almost certainly need a laywer to close the deal and if you can get a
recommendation for a good lawyer he/she should give you a free
consultation (probably 30 minutes to 1 hour)which will give a good
overview of the process.

Cheers

DPRK007