HD tubes are bulky and heavy so what I did was I took the stock tubes out to keep as spares and put the HD tubes in so that I'm less likely to even need the spares anyway. The HD tubes just take up too much space for me to keep the on hand as spares.
In addition to spare tubes, you also need a way to remove the wheel from the bike so make sure you have tools for that. For tire irons I use the Motion Pro levers or maybe the Tusk knockoffs, I can't remember which but they're basically the same. You'll also need an air compressor to reinflate the tires. It's also helpful to have a valve stem puller in order to fish the valve stem through the rim because sometimes it's a pain.
I'd also recommend practicing at home, maybe when you swap the stock tubes for HD tubes so that you know what you're up against. Everything is more difficult on the side of the trail.
Personally I don't ride my tubed bike very far from home base so I don't bother carrying tubes or tire changing tools because it'd be easier just to have my wife come pick me up or meet me with a tube/tools if I need them.
I'm probably going to give the Tubliss system a try because having to change a tube to fix a puncture sucks. A proper tubeless wheel and tire is the best option IMHO because all you need is a rope plug kit and an inflator. Tubed tires need so much extra crap that I just don't like carrying on a day trip.