On the many private school open days I've attended, most people are smart-casual or weekend casual. Think dark jeans or chinos, and a woolly jumper, shirt or polo for the men, and women in jeans/linen trousers/denim or cord skirts, with a jumper. Footwear could, weather-depending, be sandals, converse, leather boots, shoes (but not black shiny suit-type shoes IYSWIM).
I don't think I saw any lounge wear (yoga kit etc) during an open day, although anything yoga/running/jeans etc is common (as in normal) for pick up. Saying that, I've never seen anyone in 'youth' type joggers, hoodie, baseball cap and white trainers, so I would avoid that.
The only time I raised an eyebrow was when I saw an entire family dressed up in matching Burberry type jackets, country smart (tweed trousers, jackets, mustard pullover, checked shirt, tie), which if all the family were in different shades, or the colours had been more muted, or the clothes were older/battered, would have fit in ok, but the colours were all rather bright and every item looked brand spanking new (and they all matched!). They stuck out like a sore thumb - it was too bright, too try hard, too obviously expensive and new, and just very different from the other parents.
Don't worry about needing second hand uniform btw, I don't know anyone who turns their noses up at the second hand uniform shop! Everyone loves it, so there's no stigma at all about having second hand. And I know that's the same at many other private schools.