Cabs and Car Services
Your best bet might be to take a car to and from the airport. You can use a ride-hailing app like Lyft or Uber, hail a cab, or use the green Boro taxis service to prearrange a pick up in Brooklyn. You should allow a half hour to an hour for travel time; ask the dispatchers to confirm.
LaGuardia is the cheapest of the three NY area airports to get to by cab from Brooklyn. It can cost anywhere from $23 to $45 to get to the airport from downtown, depending on the time of your departure and the car service you use. If you hail a taxi, there are no tolls; however, you will need to pay for other hidden costs, including tipping, parking, traffic, and the time the cabbie spends waiting for you if you arrange a pickup. 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.
There is no subway that takes you directly to LaGuardia from Brooklyn, so you will eventually need to take a bus to get there. From Brooklyn, you can take subway lines F, M, or R to Jackson Heights-Roosevelt Avenue/74 St-Broadway. To get to Terminal A, you will need to transfer to the Q47 bus. To get to Terminals B, C, or D, hop on the Q70 LaGuardia Link SBS.
Alternatively, you could take the 2, 3, 4, 5, 6, A, B, C, or D subway lines to 125 St. You can reach all terminals when you transfer to the M60 SBS bus from there. This bus runs 24 hours a day, so you can get a ride to and from LaGuardia no matter what time your flight is.
Although taking public transportation to LaGuardia can be a huge hassle, it is the least expensive option. A one-way trip costs $2.75 when you use a MetroCard (the subway-to-bus connection is free).
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. 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 terminal parking, it's free the first 30 minutes, and $5 the first hour. It costs $5 every additional half hour you park, or $39 for 24 hours. If you are driving to pick someone up, you can park for three hours in the B Wait Area near Terminal A for free. Check the website to determine which terminal has availability.