The bad news about the meaning and use of the different Italian prepositions is that they have actually to be learnt one by one, just like any other word, taking into account that each may have more than one use.
For instance, your da in “è stato un giorno da dimenticare” pertains to one of the uses of da, to describe purposes (“proposizioni finali”): “a day to be forgotten”. Other examples of this use of da include: “una persona da ammirare” (= “to be admired”), “c'è molto da fare” and so on.
But da itself has many other meanings, from its basic one, “from”, to denote a starting point (“questo treno va da Roma a Milano”), a second basic one more or less corresponding to “by” (“un romanzo scritto da Umberto Eco”) to figurative meanings of the above, and other ones as well. Prepositions have to be learned individually, as well as their use with single verbs: sperare is used with di (“spero di vincere”), while volere requires no preposition (“voglio vincere”).
So, there is no better way than having ready a good, possibly one-language dictionary or grammar book with lots of examples, study them, but above all reading with attention good Italian texts, noticing how words are used (perhaps underlining them or taking notes or whatever works for you).
The good news is that the Italian fundamental (“semplici”) preposizioni are very few, and the actually tricky ones even less: I'd say di, a, and da, in particular.
Finally, patterns as «dimenticare + di» given in your book are not to be interpreted as you do: they do not mean you have to write something like *«un giorno di dimenticare», which is not Italian. As a commenter remarks, it is correct to say “dimenticare di fare qualcosa.” Analogously, “imparare + a” must refer to the fact that the usual construction is “imparare a fare qualcosa.” So they explain how to use that verb, i.e. which preposition is to be used after them.