Binomial nomenclature is the system of identifying organisms by their Genus species names. The full system of scientific classification, however, involves many broader levels of classification. From the broadest classification, kingdom, you move down through the progressively more restrictive classifications of phylum*, class, order, family, genus and species. A common mnemonic for learning the levels is Kings Play Chess On Fine Grained Sand. A few secondary ranks exist in the plant kingdom, but for most purposes this is good enough.
I think the best way to start learning scientific names is to go up one level from the binomial names, to the families. While there are a lot of families (more than 600!), a small number account for nearly all our common plants. Just two families, the Asteraceae (aster or daisy family) and Fabaceae (pea or legume family) account for more than 42,000 species, and nearly everyone knows what daisies and pea flowers look like. Once you've learned the characteristics of some of the common families—numbers of floral parts, type of symmetry, leaf types, fruit types—it becomes fairly easy for you to identify unfamiliar plants to the family level. Here is a great PDF you can download that illustrates 50 of the most common families. Each family has a gorgeously illustrated page listing the major characteristics of each family, along with prominent plants in each one. (For some reason, if the link won't work type “sdsu 50 plant families” into Google and click on the top hit.)
Fifty is really too many families to try to learn, but if you use that PDF in conjunction with the following link from the website Learn Plants Now, you'll be well on your way to learning about the five biggest families, which account for nearly 60,000 species. By the way, you'll notice there are often two versions of family names given—eg Asteraceae and Compositae for the sunflower family. The ones that end in –aceae are the recognized “modern” names, while the ones that end in –ae are the older ones. Most times you'll see both given, but you might as well learn the new ones.
If you prefer a book, Botany in a Day, by Thomas Elpel is a great choice. The Mono Lake Committee in Lee Vining generally has it in stock.
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* Ed. note: With plants and fungi we usually use "division" instead of phylum. Same meaning, but useful in adding unnecessary confusion. Never was explained to me why we do this. I assume it's tradition. Since 1993 it's been OK to use either. I can't remember the last time I even thought about a plant's division. Family level is more useful.