How to check my EPSG and units in R when it is not defined in proj4string?

The proj string does not contain an EPSG code. You can use EPSG:4269 or +proj=longlat +ellps=GRS80 +towgs84=0,0,0,0,0,0,0 +no_defs. Some softwares try to guess the EPSG code from the string, and sometimes they fail, and treat it as a custom CRS.

EPSG:4269 has degrees as units, and is not a projected coordinate system, but rather a geographic coordinate system. If you want a projected CRS, take the state plane systems, or a US contiguos projection. EPSG:102003 to 102005 are designed by ESRI for those tasks. You may find them in some GDAL based software, but not all.

If you want miles or kilometers, you need to put up a custom CRS. EPSG has only meters and us-feet based projections.

Well, your projection is "longlat", ellipsoid "GRS80" and datum "NAD83", so the data is unprojected and in decimal degrees. This is why is projected is returning FALSE. If you want your data to be projected you first need to choose a projection then use spTransfrom to reproject it. Since we know nothing about your data, like where it is, it is difficult to recommend a projection. This precludes a "step-by-step" solution.