You are asking about two different things: training and learning to fight. The distinction is crucial because the Spartans did one but not the other.

If you wish to learn to fight, your best bet is to move to another state. Spartans did not train weapon proficiency. The sources that explain the Spartan upbringing and training regime never mention any practice in the use of spears, swords or any other weapons. While other Greeks might be able to hire a private drillmaster (hoplomachos) to teach them some moves, these instructors were not allowed to enter Sparta. The reason was simple: Spartans were not raised to be expert fighters, but selfless instruments of their community. They needed to be brave and obedient; their skill with a spear was comparatively trivial. Like all Greek militias, the Spartan army prized courage over fighting ability, since it was courage and not skill at arms that won battles.

The only specifically warlike training that the Spartans underwent was their basic formation drill. This seems to have started when they came of age to be liable for military service (when they turned 20). Most of the work would probably be done just before a campaign or when the army was already on the march. Women could not take part in any of this because they were not liable for military service. Plato (Laws 806a-b) actually complains that, for all their healthy exercises, Spartan women are useless in war because Spartan law exempted them from anything actually related to war.

In other words, you could never join the men to learn fighting skills; the young men weren't learning how to fight, and the older men barred you from their preparations. There was apparently nothing a Spartan could or would teach you about fighting.

On the other hand, if you're looking merely to train like the men - you are already doing it. Many Greek sources remark on the unusual fact that young women at Sparta were required to exercise. In the Classical period their exercises mostly seem to have consisted of running and wrestling, but by the Hellenistic and Roman periods (after the decline of Spartan power) the training regime for young women also included throwing the discus and the javelin. In addition, women were expected to take part in various religius festivals as choral dancers, and would have had to practice dancing for much of their life.

All of these exercises were basically the same as those of the men. It was through training in these sports that men were expected to become fit and strong enough to make themselves useful as warriors - and your exercises as a young woman, at least until marriage, were exactly the same. If you wish to become a warrior like the men, you are already well on your way as far as the Greeks are concerned. All you would need to do now is find a way to familiarise yourself with wearing armour, holding a weapon and standing in formation.

You would not exercise together with the men, though, as far as we can tell. Women's training and participation in sporting events was a public spectacle as much as a eugenic measure; the men would be watching you from the sidelines. You would be watching them at their exercises in turn, and yell at them if they failed to show the proper Spartan spirit.