Cabs and Car Services
LaGuardia is the cheapest of the three NY area airports to get to by cab from Brooklyn. You can either call a car service or use the green "Boro taxis" to prearrange a pick up in Brooklyn. Allow a half hour to an hour for travel time; ask the dispatchers.
There are no tolls. Tips, the cabbie waiting time if you arrange a pickup, parking, and any other fees are not included. Good news: There are no extra fees for extra passengers.
Safety tip: Don't accept an offer of transportation from solicitors who hang around inside the terminal. They probably are unlicensed and uninsured, and soliciting at the airport is actually illegal. Look for the Port Authority Welcome Center booth, or just go outside to the taxi stand.
Traveling to LaGuardia from Brooklyn by public transportation is the least expensive option. Check out the MTA's Trip Planner for real-time travel options, with estimated times depending on the day and hour you'll be en route.
Private Express Bus & Subway Combo
You can catch a private express bus to and from LaGuardia that departs about every half hour. Going to the airport, you can get it at Manhattan's Port Authority Bus Terminal, Grand Central Station, or Penn Station. Coming from the airport, the NYC Airporter bus will drop you at these same major transportation centers.
From these hubs, you can easily pick up a subway into anywhere in Brooklyn.
Shared Ride Vans
According to the LaGuardia Airport website, "Reservations for the shared-ride van and private car services can be made at the Welcome Center located on the Arrivals level of each terminal. If the counter is closed, there is a convenient self-service kiosk nearby where you can contact authorized shared-ride or private car services." Note that these are only cost-effective if you are traveling with a group; otherwise, they will be more expensive than a cab waiting at the cab stand at the airport. Note that because they are called "shared vans" they are not particularly "green".
Still, if you are traveling with your band and a lot of gear, or going on an extended family vacation and want to go in style, it might be worth the cost. Services recommended on the airport website:
Drive and Park
Of course, you can drive to LaGuardia. There's onsite parking, as well, for a fee. Check the website to determine which terminal has availability.