Learn Italian: Volevo solo una spuntatina, per favore. - I just wanted a little trim, please.

This is a classic example of the imperfetto attenuativo (mitigating imperfect). Italians frequently use the imperfect tense to soften a present request or intention, making it sound less demanding than the direct present tense voglio (I want). It works just like saying "I was hoping for" or "I wanted" in English.

Is there a difference between using volevo and the conditional vorrei here?

Both are polite ways to ask for something, but volevo is incredibly common in casual, everyday spoken Italian. Vorrei (I would like) is slightly more formal and deferential. In a hair salon, volevo strikes the perfect conversational, friendly tone.

Break down the word spuntatina. Where does it come from?

It stems from the noun punta (point or tip). The verb spuntare means to cut the ends off, or "to trim". From that, we get the noun spuntata (a trim), and adding the diminutive suffix -ina turns it into una spuntatina (a little trim).

Could I place solo at the beginning of the sentence instead?

It is much more natural to place adverbs like solo (just/only) immediately after the conjugated verb they modify. Saying Volevo solo... flows perfectly in Italian, whereas Solo volevo... sounds clunky and unnatural.