Ok - as a note - saying the camera is compatible with Mac, by most of these companies it seems, is meaning it's compatible with iMovie.
You can open iMovie, connect the camera, have iMovie import the footage and then use it to convert it to another format (you can drag the imported movie into the edit area and then share the movie) - DO NOT attempt to import the individual .mts files into iMovie, THEY WILL NOT WORK, you need to use iMovies functionality to allow it to convert the AVCHD footage of the camera into something more Mac friendly.
Other options that may work for transcoding include:
MPEGStreamClip can convert some .ts and .mts files into something playable on a Mac (free)
VoltaicHD can convert .ts, .mts and .m2ts files pretty well, but IS REALLY REALLY REALLY slow, and costs money
Handbrake - works with some .ts and .mts files, they are improving their HD footage support (as well as the associated stream containers, this has been what I've been using lately to downconvert some 1080p footage to 720p that was in either a .ts or .m2ts container), free
I've heard you can convert some .ts and .mts files with Roxio Toast as long as you have the HD plugin, but you're talking ~$100 for Toast and the plugin
Players:
Mac doesn't really have any nice AVCHD players that will play by just pointing the player to the AVCHD folder, you will need to navigate to the specific .ts, .mts or .m2ts files and play. Players that tend to work ok with those file containers:
vlc (which you already have)
mplayer for mac
Plexapp
In reality, there just isn't really good AVCHD support in the Mac world - there are many more apps PC side to deal with AVCHD although I expect this will change. Your probably easiest options are to:
Try handbrake - it works really well, I use it quite often and have started using it with some HD footage. Hasn't worked with all, but it has worked with enough that I use it often.
Use the iMovie trick of ingesting the video into iMovie then export it using the share function into a file format that is Mac friendly.
From my AVCHD camcorder, if I'm not editing, then I ingest the footage and then export it to what I want (I use Final Cut, but you can do it with iMovie as well). Either that or I convert it using handbrake into something more Mac friendly (like an mp4 or mkv container H.264 video).