The art of fighting with hoplite equipment was called hoplomachia, literally "armed combat". The Classical Greeks generally seem to have considered this art unnecessary and undesirable for reasons that are not important to your question, but at least they acknowledged that it was a skill that could be taught and learned. In one of his dialogues, Plato has the experienced general Nikias explain the advantages of hoplomachia:

This accomplishment will be of some benefit also in actual battle, when it comes to fighting in line with a number of other men; but its greatest advantage will be felt when the ranks are broken, and you find you must fight man to man, either in pursuing someone who is trying to beat off your attack, or in retreating yourself and beating off the attack of another. Whoever possessed this accomplishment could come to no harm so long as he had but one to deal with, nor yet, perhaps, if he had several; it would give him an advantage in any situation.

-- Plato, Laches 182a-b

Nikias specifically points out that learning hoplomachia will prove beneficial in situations where hoplites have to engage in single combat, such as during a rout. When a phalanx broke, it would typically be chased by the enemy's hoplites first, but when these ran out of steam, light-armed troops and cavalry would take over. The scenario sketched by Nikias therefore implies a range of different opponents. As he underlines, learning the skill of using hoplite weapons will give the wielder an advantage every time.

Scholars have traditionally assumed that the hoplite shield was only good for fighting in formation. It is bulky and strangely shaped so that every man's shield extends to his left to cover the man next to him. But the passage above shows that this is not how the ancients saw it. Even if the hoplite's primary function in the time of Plato (4th century BC) was to fight in formation, it was perfectly obvious that this wouldn't always be what happened, and so hoplites also had to be prepared to fight alone. Indeed, there is no solid evidence of the existence of the phalanx for centuries after hoplite equipment was introduced (in the early Archaic period, c.720 BC). Apparently, this type of equipment was originally intended for single combat, nor for fighting in a not-yet-existent formation. I wrote about what this sort of combat might have look like here.

What I would worry about, though, is whether hoplite equipment would be convenient for a wandering warrior. The shield in particular is heavy (4-8kg) and awkwardly shaped; Greek hoplites tended to have their slave carry it while they were travelling. If your character is travelling alone, she might be unnecessarily burdened by this heavy item. Better to go for a smaller round shield to pair with her spear; there are plenty of types available from all over the ancient world that would be less bulky than a hoplite's aspis.

The spear and shield combination of hoplites was used as part of a group. Often the shields were in defense of other spears, but I understand what you mean. There is such a wide range of spears, shields, and swords though (along with their styles and not to mention other weapons) throughout history that it really depends on what exactly you are picturing. Are you curious mostly about what would have been possible in an ancient Greek setting, or in some other period or place?

The closest I can think of that you'll find for actual examples of this happening, and the effectiveness of it, may be the ko-naginata. It's more a pole weapon than a true spear, but was used extensively throughout the Edo period by women in Samurai families. It had a lighter weight than the version used by men, and had a longer reach that could help a smaller person against a larger one armed with a sword. This is not ancient, but may help your research, and information is readily available if you do find it useful.