Keyword: cofinality.
http://en.wikipedia.org/wiki/Cofinality
Added MUCH later: To be slightly more explicit, for any cardinal $\kappa$, the cofinality of the successor cardinal $\kappa^+$ is $\kappa^+$, so not only can we not in general find an unbounded (=cofinal) countable subset, there is no fixed cardinality $\kappa$ such that every totally ordered set has an unbounded subset of cardinality at most $\kappa$.