I'm a complete beginner who will be interning in Seattle this summer and I'm looking to do some climbs of mountains like Mt. Hood, Mt. Adams, and (very unlikely) Rainier. I've looked at the gear lists on other websites for alpine climbing and there seems to be a lot to get. However, as I'm only going to be here for the summer, I'm wondering what layers are essential to buy. I've already got some gear from backpacking: base layer(Patagonia Capilene 1), light fleece(R1 Jacket), and a rain jacket(Patagonia Torrentshell).
I've still got to get some bottom layers, but I'm wondering what else is absolutely necessary for summer climbs. I probably need to get a insulated jacket, but I've got a down jacket from Uniqlo which I don't know is good enough or not. I also am not sure whether I should get a hard shell or not, considering I already possess a rain jacket. If anyone experienced with gear could shed some light, that would be much appreciated.