BY AIR:
The international airport in Monterrey is called General Mariano Escobedo Airport.  It is located to the north of the city, and probably takes about 20-30 minutes to get downtown when traffic is mild.  The airport is very small if you're used to something like Houston or LAX, but there are all the basic amenities there.  Duty free shops, restaurants (both inside and outside the terminal), ATMs, etc.

Many airlines fly into Monterrey.  I know Continental and American are regulars.  Aviacsa is a Mexican airline that has a few direct flights between some US cities (LA, Chicago, Las Vegas).  There are also Mexicana, AeroMexico, and a few others.  MagniCharters is a tour company that often does package trips from Monterrey.  I know you can buy tickets to fly on their plane to some of their destinations, as well. 

 I don't believe there is a departure tax that is separate from anything that is included in your ticket.  I've never had to pay anything like that. If you are leaving Monterrey and hold an FM3 or FM2 visa, it is becoming more and more strict that you make sure your little booklet is stamped when you leave the country, or you could get hassled/fined when you come back. This rule used to be pretty relaxed, but the immigrations officials have been getting pretty strict lately.  I've even heard this about driving across the border these days (it wasn't like this before).  Be sure to ask about it.  If you are a tourist, I don't think you have to worry about it.

 When you arrive, at least in the domesic baggage area, there a few hotel desks and car rental counters.  You have to actually exit the baggage area to arrange a taxi.  When you do exit the baggage area, the taxi salespeople will be leaning over their counter, offering their services to you.  There are set prices based on the region you are going to.  It can cost about MXN$200 (give or take) to leave the airport, and it is slightly cheaper to take a taxi to the airport from town.  There are several taxi companies for you to choose from.  I am sure they are all reputable. 

 At peak traffic times, plan on a lot more time to get to and from the airport.  Traffic gets crazy.  I believe 5-8pm is pretty heavy on weekdays, but I don't know much about mornings.  Sundays are by far a very quiet day on the roads.  There is also a new toll road (I think it's about MXN$20) that you can take to the airport from Constitution.  If there's no traffic, you really don't need to use it, but if you're travelling at peak time, you should consider it.

BY CAR:

The main ways into Monterrey are via the E-W Highway between Saltillo and Reynosa (McAllen TX border).  You can take a toll road if you're coming from either of these places, but it bypasses the city.  Though there's not a major "freeway" (per se) through town, you'll basically be driving the length of Constitution.  Constitution goes E-W and takes you right past downtown.  There tends to be a lot of traffic near downtown, so be sure to avoid peak commuting time.

 If you're coming from Laredo, the road takes you straight into downtown.  Again, you will hit a lot of traffic unless you plan your trip around it.  Depending on where you're going, you can find a quicker way to get where you need to.  To go downtown, however, you'd basically go "straight" into it.

Coming from the south, you'll probably be on the road from Victoria, or on the highway from Mexico City.  The highway from Mexico City is actually via Saltillo, so you'd enter Monterrey from the west, and it's fairly straightforward (puts you out by Constitution).  If you come in from Ciudad Victoria, the highway takes you past Santiago and the lake/presa.  You'd be entering Monterrey on the east end of the city, along the Saddleback Mountain.  On holidays, this highway can get heavily backed up.  Keep this in mind when planning your travel times.  We spent a good two hours just sitting in traffic by the lake on a holiday in the late afternoon.