there are lots of ways to summarize and visualize this data. personally, i like the way BONAP gives you a bunch of species-level maps all together for a particular genus (ex. http://bonap.net/NAPA/TaxonMaps/Genus/County/Ambrosia). which tool you use depends on your audience and your skill level.
if you’re just making stuff for yourself, it’s probably easiest for most people to do this in a program like QGIS. if you want to get to state level and you don’t already have that associated with your coordinates, you can do a spatial join between your point data and a set of state polygons. from there, you could summarize that data as just a distinct species list per state and/or you could make a choropleth that shows species count by state. you can search the web for how to do most of this. this other thread (https://forum.inaturalist.org/t/self-made-distribution-maps-in-qgis/33891) might also be helpful, as it covers a few things that sound similar to what you’re trying to do.
if you want to make this for a larger audience, then it’s probably best to have some sort of web-based presentation of the data. i think any way you approach this, you’re going to have do some amount of coding to achieve what you’re describing. so i guess how you go about it just depends on how comfortable you are with different coding languages, etc.