I have a standard container box under the bed with all my sport gear in it, covering summer and winter (in so much as such a thing exists in Northern California, though I could definitely survive an Irish winter with what I have, and my impression is that Portland has a similar climate to Dublin), cycling, rugby, and lately, running. There's a bit of overlap between the 3 sports, such as baselayers.

If not breaking the bank is important, I can definitely recommend the Champion baselayers and gloves at Target. I have 2 long sleeve, 1 short sleeve undershirts that have lasted me at least 5 seasons of rugby training, so you know they stand up to wear and tear. The running gloves are just the right weight for cycling in winters here, and have finger pads that allow you to use touch-screen devices like my phone without taking the gloves off. I have a headband that covers my ears on particularly cold days; my hair is thick enough that I don't need/can't fit a hat on my head under my helmet.

I have a thermal base layer by Canterbury that I bought for rugby in Ireland but never wear any more because it doesn't get cold enough here.

Some things you need to get the cycling-specific gear; Get a good pair of bike shorts. It's worth spending money on the shorts, but it's also worth looking for reduced prices on top quality stuff on discount racks in REI, Black Friday sales etc. Overshoes/toe covers are better for keeping your feet warm than thick socks.