You do not want to take a bus-only trip from La Guardia to Times Square; it would take too long.  You will want to transfer to the subway.

I don't know about buying MetroCards at La Guardia, since I live in New York.  Many years ago, I did once try to buy or refill one there, without success.  I ended up taking a taxi (at the time I lived close to the airport, so it was maybe a $10 or $15 ride).

Still, with two young children, I would be very strongly inclined to take a taxi.  However, if you're not in a terrible hurry and if it's not too crowded, you might be okay with the bus.  Because you'll need to change to the subway, you'll want to be sure that you don't have too much luggage to move all at once.

The MTA has a [brochure](http://web.mta.info/nyct/service/NewQ70LimitedStopService_brochure.htm) promoting the Q70 limited bus service, which connects with several subway lines serving midtown Manhattan.  This brochure indicates that there are now MetroCard vending machines in the airport:

> You can buy a MetroCard that gives you unlimited rides (that includes all free transfers). Or, you can buy a MetroCard with a specific number of rides that automatically gives you one free transfer every time you pay a
fare. You can purchase MetroCard at MetroCard Vending Machines inside Terminals B, C, and D in LaGuardia Airport or at any NYC subway station.

"One free transfer" includes transfers between buses, or a transfer from bus to subway (or vice versa).

My experience is mostly with the M60 bus, which goes to upper Manhattan, because it passes by where I used to live.  It might help you if you're going close to the Museum of Natural History.  The M60 has changed to a "Select Bus Service" line since I last used it, so the buses are longer (articulated) and the crowding problems may have abated somewhat as a result.

There are a few other possibilities, too; I usually use Google Maps to plan my rides on public transit, but you'll also want to check the [MTA site](http://www.mta.info), especially if you're traveling on the weekend.  There are frequent service disruptions because of construction activities.

The Q70 Limited brochure indicates that these buses are designed to carry baggage, contrary to my (not-so-recent) experience with the M60:

> To accommodate airline passengers, the Q70 LTD is outfitted with special luggage racks, so you can stow your bags as you travel on our comfortable, climate-controlled, handicapped accessible buses.