For schooling shows you should be able any sort of safe, certified helmet (which tipperary are) in good condition. The only time I'd suggest getting a different helmet if you're just doing schooling shows (for now) is if you have one that's pattered or a bright/obnoxious colour, but even then it shouldn't effect how you are judged. I have a grey tipperary helmet and show in it all the time, and have shown in recognized (low level) dressage classes with it before. I even had a hunter green one before that I took in recognized dressage classes. One day you might want to invest in a show helmet that's a little more trendy in terms of the discipline you choose, but I wouldn't worry about it until the time comes.
As for footwear, most schooling shows just require boots with a heel, and half chaps are usually just fine. Just make sure you read the schooling show rules to see if it's specified. However for bigger shows tall boots are usually mandatory. But even then you will have to check the specific show rules, and the rules of the organization putting on the show. For instance, sometimes rubber boots are still ok, they just have to be tall boots. Some times they must be leather. Sometimes if you're showing dressage you must ride in leather dressage boots and leather field boots aren't even allowed.
But the bottom line is, for schooling shows, what you have should be fine, but read the specific show rules. Don't worry about getting new stuff until the time comes that you're ready for bigger shows.