Learn Russian: Я хотел купить билеты десятого апреля две тысячи пятнадцатого года, но забыл. - I wanted to buy tickets on the tenth of April two thousand fifteen, but forgot.

To say on a specific full date, Russian puts the ordinal number for the day into the Genitive case. That is why we use the Genitive десятого instead of the Nominative десятое. You don't need a preposition like "on" at all!

Why didn't две тысячи change its ending to match the Genitive case of the year?

When expressing a year in Russian, you use a compound ordinal number (literally "the two thousand fifteenth year"). The rule for compound ordinal numbers is that only the very last word declines! So две тысячи stays in its base form, and only пятнадцатый changes to the Genitive пятнадцатого.

Why do десятого and пятнадцатого end in "-ого", and how is that pronounced?

Ordinal numbers decline exactly like adjectives. Since they are in the masculine/neuter Genitive case here, they take the -ого ending. Remember that in these adjective endings, the Russian г is pronounced like a "v" (so десятого sounds like "de-SYA-ta-va").

Why do we use the perfective verbs купить and забыл, but the imperfective verb хотел?

хотел (wanted) describes an ongoing state of mind in the past, which requires the imperfective aspect. However, the action you wanted to successfully complete (купить) and the one-time event of forgetting (забыл) are viewed as finished results, so they use the perfective aspect.