Agreed that playing Halo 4 would be smart. ODST was a waste of time, IMO. I played it a month or so ago with a friend. I still don't know what the heck was going on, nor do I get why people find it to be a good campaign. Seems like the hipster title where people just say "I like it because it's different," even though it's an aimless love story, disjointed progress, and little content.
Watched through it and enjoyed it personally, the current events with Rookie walking through interspersed with flashbacks of his team, eventually linking up with them afterwards and continuing a solid story in the final act.
Great atmosphere, I'm actually working to ODST's music right now (Asphalt and Ablution) etc.
But then again Beyond: Two Souls and with its events being all jumbled up chronologically made sense to me so...
But considering Buck is in Fireteam Osiris, I thought it'd be a nice background thing to know.
But yes, there's a lot that goes down between 3 and 5, going through 4 and Spartan Ops would definitely be worth a go to minimize being lost. (TBH Halo 5 felt like some things would be lost on someone who just played the core games like characters and small details)