Best way to get to the airports would be by car. However, going by the metro would be the cheapest alternative, but its also VERY confusing finding the right stations in both Moscow and Petersburg, not to mention the trouble of tagging your luggage along. If you really want to go by metro, you must figure out the route in advance, and make sure where exactly you're going. If you prefer using the car, best way would be to arrange private transport for the sake of timeliness and possibly costs. Cabs usually charge anywhere from $38 to $50 or more to get to the airports from downtown in both towns, though I also heard that the government wants to clean up the act and lower prices for Moscow. Maybe your friend can also arrange a private vehicle for you in Moscow, and that would be best. If not, you can check out the express buses which go downtown from the airport, and they're cheaper than the cabs anyway. The other thing to take note of is traffic jams and timing. Moscow is HUGE so expect a long drive. Petersburg isn't that big but traffic jams along the southern parts are very common, and since Pulkovo airport is in the south, you might want to take note of that when arranging your departure flights. For the trains, just remember that for the first hour or so going into and out of both towns, they lock the toilets up, and no pleas in the world would make them open it. They call it the "zone", though I have no idea what good it does.
Only good thing is that the trains land you downtown in Petersburg, near Vasstaniya Ploshad, so its pretty convienient. Hope this helps.
