when is it "da" and "di" before infinites? Sometimes I think I see "a" too?
In Italian it's very rare to find two verbs without a preposition connecting them. It usually only happens with modal (volere, potere, sapere, dovere) verbs.
Everywhere else they're normally connected by a preposition, most commonly a or di.
The former typically is used when the first verb has the second as a goal or intention e.g. "I asked you (a favour in order for you to/so that you would/because I wanted you to go" or Ho cercato di venire = I tried (my best in order to/to do all I could so that I could) come. It is also used for claims like L'insegnante dice di studiare = The teacher said (something claiming) to study.
In the latter the second verb has more of a spatial destination (it's most commonly used with all verbs of motion like go, come, run etc.) so e.g. Vengo a mangiare = "I come (to the place where I can) eat" or Tornate a dormire! = "go back to sleep (i.e. to the place where you can sleep)"
Interessarsi (to grow an interest) and essere interessato (to be interested) as well as iniziare (start) and cominciare (begin) fall under this second catgory.
But notice there are exceptions! The first I can think of is the verb provare which always takes a instead of di (but its synonyms cercare and tentare take di as you can see in the example above).
As far as da is concerned it only usually connects nouns to verbs, e.g. Ho molte libri da leggere = "I have many books to read (i.e. which I must read)" or Non c'è nulla da fare = "There is nothing to do (i.e. that can be done)". But notice it can sometimes go on its own after the verb to have, e.g. Ho da rientrare presto stasera = "I have to go home early tonight". In this case it actually equated to the form "ho bisogno di" = I need.
Hope this helped you to get at least a general idea of how verbs are connected in Italian ;)
'We have a lot to win.' should be accepted see: https://www.collinsdictionary.com/dictionary/italian-english/guadagnare and https://en.wiktionary.org/wiki/guadagnare
Can someone please help? I'm having trouble understanding the rule for DA. I know it expresses some kind of purpose or end but how do you distinguish that from the others????????