... And yes, rationally connected varieties are uniruled. This should be obvious: the standard definition of rational connectivity is that there is a dominant map from $T \times \mathbb{P}^1 \times \mathbb{P^1}$ induced diagonally from a morphism from $T \times \mathbb{P}^1$.

PS: I realized I had misread your question. Certainly uniruled varieties need not be rationally connected: e.g., a ruled surface $C \times \mathbb{P}^1$, where $C$ is a curve of positive genus, is not rationally connected as it has non-zero holomorphic $1$-forms (coming from $C$).

For surfaces, rational connectivity coincides with rationality. In general, it is even an open problem whether there are any rationality connected varieties which are not unirational. This is called an "embarrassing question" in notes by J. Kock on lectures by Joe Harris: mat.uab.es/~kock/RLN/rcv.pdf . For a recent confirmation that this is still an open problem, see Jason Starr's answer here: mathoverflow.net/questions/131086/… .