I've had good luck with strollers from brands like Valco and Baby Jumper. They tend to have spacious under-seat baskets, especially when you consider their overall size and foldability. For instance, the Valco Stroller Trek has a decent-sized basket that can hold quite a bit.
As for other features to look out for in this budget range, I'd recommend prioritizing something with a good canopy or UV protection. That way, baby will stay protected from the sun's rays when you're out and about. Also, consider a stroller with a sturdy, adjustable handlebar that fits comfortably against your shoulder or underarm.
In terms of essentials to keep in the under-seat basket, it depends on what you need to carry but some common items include extra diapers, formula or breastmilk , wipes, pacifiers , and a few toys or books for soothing or distraction purposes. Some parents also prefer keeping an extra change of clothes or a small blanket just in case of accidents or spills. What's worked best for you in this regard?