Does anybody know the best way to get from Park Slope to the Upper West Side (and vice versa)?
I prefer to take the Manhattan Bridge but I was just wondering what streets other people prefer to get to either bridge...

Probably not the best way, but I take the Brooklyn Bridge, then go across town on Chambers and up the Manhattan Greenway. It is a pain to dodge tourists on the Brooklyn Bridge but I try to leave enough time to enjoy the view (having a bell helps). To get to the bridge from Park Slope, I take 4th or 3rd Avenue to Bergen Street, Smith Street, then either Jay Street to Tillary to the bridge, or Schermerhorn to Boerum Place to the bridge. The Manhattan Bridge is much more peaceful (no, or very few pedestrians!) however I use it primarily when my destination is on the East Side. Are you familiar with Ride the City, where you can map point-to-point routes? Google has a similar route planner, I believe.

I take Vanderbilt to Flushing Ave at the end, and then Navy and the bike lane to the Manhattan Bridge. It is bike route the entire way, although Vanderbilt only has a bike lane to Fulton and then is a "shared" roadway. There still isn't that much traffic to deal with.

The problem is the Manhattan Bridge leaves you on the east side. That's OK for me as I work at 2nd Ave, but to get to the UWS you've got to cut across. At least one crosstown street is marked somewhere south of Houston.

I'll leave the Brooklyn section to folks who live in Brooklyn.

My current route to get from the Manhattan Bridge to the west side is to just continue along Canal Street, across the Bowery, to Elizabeth. Then turn west on Prince Street and from there on it's bike lanes. Follow Prince till it turns into Charlton. Then take a quick right on Greenwich Street, left on Houston for two blocks where you can catch the West Side Greenway.

Traffic gets pretty thick on Elizabeth but I haven't found a better street in that area between the bridges. The upside is it's a through street, with mostly passenger cars, and you don't have to dog leg around the Bowery or ride against traffic.

There's a more-or-less marked route that takes you uptown on Chrystie to Rivington for a block on the Bowery to Prince. Prince becomes Charlton as you say which you take all the way until it ends whereupon you make a right on Greenwich up to Houston. Much safer and smoother since Charlton was repaved, IMO, though the bike lane on Prince is some kind of sick practical joke on cyclists (I swear it was safer to ride there w/o it).

As for the Brooklyn bit, from R&A (a LBS, FYI) the other day I rode up 5th Ave to Bergen to Smith which becomes Jay St which goes to the Manhattan Bridge. Jay becomes extremely hairy for the last block before the bridge but it's otherwise pretty good and mostly bike lane / "sharrow".

Yeah that's the alternate route. I haven't done it in a while but I think DOT made some improvements to the Bowery & Prince intersection. Maybe they even did away with the dogleg.

Well, as to the Brooklyn part, this is a route I ride often:

From President and Prospect Park West:
Down President,
Rt on 3rd Ave
L on Bergen for just 2 blocks
R Bond St
L Pacific St
R Smith
L Schermerhorn and then I continue on towards Brooklyn Heights. I think you could go right on Boerum Place which'll take you up past the Marriott and towards the bridges. As mentioned above, you can get there by Smith.

A nice alternative to the problem with Jay St. is, if you continue on that little road next to the Brooklyn Bridge entrance (I think it's Adams), past the apartment towers, it hooks to your right and meets up with the Manh. Bdge, much less traffic this way. My route is a crazy zig-zag that mostly avoids traffic, although 3rd can be a bit of an adventure.

One of these days I have to try the zacster route.

And I will certainly try Stacy's route too because my western routes from Manh Bdge have been unsatisfactory.

You really should try it. Every time I go through Carroll Gardens/Cobble Hill I regret it now. Ft. Greene is really nice and you avoid the worst of the traffic around the bridges.