There are some good examples of doing this over at the R Wiki. I'll steal a couple here:
Merge Method
Since your keys are named the same the short way to do an inner join is merge():
merge(df1, df2)
a full inner join (all records from both tables) can be created with the "all" keyword:
merge(df1, df2, all=TRUE)
a left outer join of df1 and df2:
merge(df1, df2, all.x=TRUE)
a right outer join of df1 and df2:
merge(df1, df2, all.y=TRUE)
you can flip 'em, slap 'em and rub 'em down to get the other two outer joins you asked about :)
Subscript Method
A left outer join with df1 on the left using a subscript method would be:
df1[,"State"]<-df2[df1[ ,"Product"], "State"]
The other combination of outer joins can be created by mungling the left outer join subscript example. (yeah, I know that's the equivalent of saying "I'll leave it as an exercise for the reader...")