There's not really a super secret way to get better fast. Just set goals for yourself and work on a couple simple things at a time until you perfect them.
At stick and puck, maybe spend the first 20 minutes just skating lines by yourself. Go at your own pace but push yourself. Maybe throw in a stop and start facing your weak direction. Or every 3 lines, skate one length backwards etc etc. During your stick and puck games, don't just use your arsenal of good things, since your developing, at practice (stick and puck) is the best time to start trying your weaker points in game situations.
Also try and play with people similar level to yourself, or just a little bit above your level. If you play with people much younger/worse you will develop bad habits and not put the effort needed in. If you play with people that are a huge amount better, you won't get the practice in because you won't be able to keep up.
I wouldn't go to a power skating coach just yet. Make sure you have all the basics down pat first, like backwards crossovers to the opposite side, and stopping the other way as well. As for instructors that don't require full gear, contact a private one, it will cost a little bit more, but he will come to stick and puck with you and work alone on what needs improvement. If you go to a large class, usually they just hand out universal tips, and only give individual help rarely. A private instructor will adapt to your needs and help you with what you need the most work on.