I'm a FCP user now migrating to MC. Among many things, not being able to move the position indicator by increments of 1 second really bothers me. On Final Cut Pro, I can do that by holding down the shift key then press left/right arrow once to move the playhead by one second. Can anyone tell me if MC also has a neat keyboard command like that on FCP?

I don't know how "neat" it is, but you can advance any number of frames (or time) forward or backward by using the number keypad. 

With the Source or Record monitor active, you can type +30 (in a 30i sequence) or +100 for 1 second (it automatically will put the colons to seperate seconds from frames etc)

Or you can press the Forward 10 frames button 3 times (in a 30i sequence)

Then just hit "Enter" to advance that same amount -just one keypress.

And following Kenton's advice about the numeric keypad, the period (.) key there functions as a double zero.  Once you have entered the value you want as itmatters says, repeatedly hitting the "Enter" key on the numeric keypad will advance you by the same amount.  The same function applies in trim mode too.  If you're more comfortable working in frames, press the F key before pressing Enter.

But if I can suggest, make the effort to not drive Avid the way that you would FCP.  It isn't, and trying to force it to be will only slow you down.  In particular, get to know the command palette and where those commands are mapped on your keyboard.  Familiarise yourself with the different workspaces.  Avid is a very powerful tool if you let it be.