It may be controversial but I'm not sure trainings and conferences are the best way, but it definitely is the way which requires least effort from your PM.
First of all, you're in a pretty good position to learn about project management. You are a project leader so I suppose there are some tasks you're fully responsible for - you have to deal tasks, coordinate people and generally ensure the job is getting done. That's basically what PMs do but in a larger scale.
Second, something you'd probably make some good use of is full responsibility for some project. If that isn't possible in your current workplace look for other opportunities. The most obvious one is organizing an event for your friends (like vacations or a trip). Also you may consider joining some voluntary project which is happening around. Although they aren't very typical project you can learn a lot being a part of organizers of such events.
Three, I would consider joining other organization. I don't want to be unfair as I base on limited information you shared in a question, but think whether people in your workplace would support your chosen career path. Also that depends of course on project but at least some of projects done for European Union don't look like typical commercial projects so they may not be the place to learn.
And four, which is connected with the previous point, if you think about changing workplace I find small companies great places to learn. Not only is learning curve very narrow but also, pretty often, it's much easier to be promoted as the company grows.
