I think shoes belong in your car or at home. I think you will find that skating with a pack large enough to hold normal shoes is awkward - at least it is for me on the ice. If you must - I suggest a lightweight pack that has straps on top and bottom that will hold the pack tight against your body. I currently walk and jog with a small fanny pack that also has two water bottle carriers, for quick access to the water, that is adequate for everything I need. You may want a ziplock baggie, for your phone (and car key if it has a transponder), in case it rains, though my fanny pack is sufficiently waterproof without it. You could use a waterproof bag with a feed through for headphones, if you need that.
Could you walk to your destination wearing skates using wheel covers? I've not tried that, because I can inline from my home. But for ice, I often use plastic blade covers to get to the rink.
Another option, assuming the reason shoes are a problem is that you have to go through gravel to get to where you skate - get off-road inlines, which you can keep on over gravel.
Or get cheap super-lightweight compact flip flops or watershoes you can walk over gravel, then stick in a small fanny pack or a coat pocket. You can probably buy the shoes for about a dollar at Walmart. At ice rinks, I usually bring cheap shoes I don't care about, and leave them under a bench.
In every other respect, outfit for inline the way you would for any other outdoor sport, where you may go hours away from home or car. Get top quality durable outdoor gear. If you do a lot of outdoor sports, top quality gear is worth it. What you carry is life-and-death survival gear, and must work.
I love coats with zippered pockets, for all sports when it isn't super-warm. Great for your phone and keys, and if you get the compact shoes I just mentioned, they could fit too. If the pockets are big enough, water bottles will fit. So will a Covid-19 cloth mask, in case it gets crowded. While you are at it, carry a first aid kit (cloth and paper tape rolls folded flat will do if you know what you are doing), a compass (I keep one on my watchband), sunglasses if they might be needed, some cash, a local map, and anything else you might need. Occasionally outside I carry a small roll of duct tape folded flat for emergency repairs, but that is probably overkill. OTOH, a tool to retighten your wheels if they come loose just might be useful - probably a suitable hex key.
In the early 80's I bought an Open Air Wear breathable and washable pile ski jacket whose entire front had big pockets, with lots of space. I often use it instead of a pack. I love it so much that when the pocket liners started to go bad, I had them re-sewn, and it's still my favorite jacket. Like a lot of high quality coats, it has long "pit zips" under the arms you can open up for ventilation when it gets a little warmer. When it gets really cold, I wear something over it that cuts the wind, like a good packable raincoat, that easily fits in one of the pockets.
I guess a good seamstress could add big zippered pockets inside almost any coat, but don't go too cheap on the coat - in addition to what I said about top quality gear, it has to take falls.
On the cheap, you could try a fishing vest, but it looks dorky, and will be warm. I tried one for backpacking and hated it.
I sometimes carry a phone and keys in a zippered wrist strap. (Note: the cheap ones aren't durable.) Or in a neckstrap mounted secure travel bag, or in a small fanny pack. But honestly, I usually leave my cell phone home, unless I think I might get lost, and need it for navigation. It's fragile, and might break in a fall.