Well, if you’re looking to get started on botany after moving to a new continent, sedges might be the wrong end of the stick to grasp. Nevertheless. If you’re seeing sedges blooming now, they’re probably Carex sect. Acrocystis or sect.Careyanae, which start blooming with the early spring ephemerals, earlier than most sedges. (Compare https://www.inaturalist.org/observations/41037432 caught on Sunday.) If rhizomatously spreading, it’s probably C. pensylvanica but could also be C. lucorum; sedges are generally best IDd when in fruit, to make use of perigynium morphology.
OK, that was a bit much. For easing into botany in the mid-Atlantic, I’d recommend Newcomb’s wildflower guide. The scientific names have not kept up with the past ~35 years of taxonomic change, and it’s not completely comprehensive, but it has a great system of keying: generate a 3-digit code based on features of flower and leaf, go to the appropriate page range, and then key dichotomously until you reach your species. The language is mostly non-technical, so it gets you used to keying in the field without the frustration of a technical flora.
I don’t know how much of a field season we’ll get this year, but one thing that really helped me become a decent field botanist was going on trips with local botanical clubs. I typically run with Philadelphia Botanical Club, which has a lot of activity in the Pine Barrens, but you’re right in the heart of the Torrey Botanical Club’s range–we did a joint trip with them to the Sourlands about a year ago, where you get the same kind of diabase-and-shale habitats that I assume you have in the Watchungs. I’d recommend a mix of roaming round on your own hook armed with a Newcomb’s guide, combined with participation in club trips/nature walks. There’s usually a broad range of expertise, the experts are really friendly and happy to explain how they make their determinations or to answer what you may feel are foolish questions, and gradually a lot of knowledge rubs off. You also start getting a feel for how the same species appears in different seasons. Often, you start by recognizing a plant when it’s in flower, and after repeated exposures, you start noticing the foliage even when it’s not flowering. You might also appreciate https://botanydepot.com/2018/01/28/50tempmanual/ which will help you identify unknown plants in the mid-Atlantic to family; being able to jump to family or genus on sight lets you bypass a lot of painful keying-out.
When you’re starting to feel comfortable with that, and you’re ready to tackle some bigger challenges, like grasses, sedges, etc., you’ll probably want a technical flora as a backstop. I don’t think there’s any recent flora of New Jersey specifically. Gleason & Cronquist is comprehensive, but getting old, and by current standards probably lumps taxa excessively. Rob Naczi at NYBG is leading the effort to replace it; you can buy individual PDFs of the families completed so far, but there’s quite a bit left to do. Rhoads & Block’s “Plants of Pennsylvania” would probably work pretty well for northern New Jersey.