I usually start out with some combination of:
typeof(obj)
class(obj)
sapply(obj, class)
sapply(obj, attributes)
attributes(obj)
names(obj)
as appropriate based on what's revealed. For example, try with:
obj <- data.frame(a=1:26, b=letters)
obj <- list(a=1:26, b=letters, c=list(d=1:26, e=letters))
data(cars)
obj <- lm(dist ~ speed, data=cars)
..etc.
If obj is an S3 or S4 object, you can also try methods or showMethods, showClass, etc. Patrick Burns' R Inferno has a pretty good section on this (sec #7).
EDIT: Dirk and Hadley mention str(obj) in their answers. It really is much better than any of the above for a quick and even detailed peek into an object.