I am a high school student who became a Christian in February and I feel like I could give you guys some useful advice.
I became a Christian largely of my friendships with friends who are Christians.
I do not want to offend anyone who is a youth ministry volunteer/ worker. I volunteer in the children's ministry at my church so I understand the struggles that us volunteers go through.
One of the two best ways to reach the next generation is to teach the Christians in that generation how to reach people [how to lead people to Christ] and let them do the witnessing to the next generation. It worked with me and I have heard countless stories of this working.
The second of the best ways is for the volunteers/workers/pastors build relationships with the youth first. Let the volunteers get to know the youth, and have them ask questions (general at first, more specific ones later) about the youths' lives (let the youth ask the volunteers questions if they wish). Let the youth guide the direction of the relationship, conversations, and questions as they all have different needs, questions, personalities, etc. Some youth might ask spiritual questions more quickly, others might need more time to see what it is about. They will likely eventually ask spiritual questions once the relationship bond is strong. If they do not, the best way to facilitate this is to slowly, gradually, and subtly over time ask them questions about where they are spiritually. If they are resistant or skeptical , continue the relationship to where it was before and scale back the spiritual questions and ask them again when they get comfortable. They will eventually come around to either answering them or asking them; us young people have been lied to by and hurt by the world more than previous generations- so we tend to be more skeptical about things. If they are receptive, let them dictate the pace of the spiritual discussions/questions. Recruit volunteers who love people, especially youth, that are good listeners, compassionate, give good advice, have personal relationships with Jesus, who live as much as they can by what they teach, and that have wisdom.
In a nutshell, teach the youth how to lead people to Jesus and let them do it and/or have qualified Christians build relationships with the youth (conversions will eventually come either way). There are some other good ways but I don't know of any more methods other than these.
By the way, I have considered volunteering in youth ministry as I get older.