Alright two key things to keep in mind: Are you going solo or you grouping with friends/public.
Both have pros and cons and it's entirely up to you. I personally like having my buddy watching my back in some games when we're tagging mobs fast. It also provides multiple abilities from different characters which can benefit each other and is invaluable in hardcore.
Next is to choose your class. Pick this based on your playstyle. If you know you're sloppy with reaction time and dodging nukes it may be best to avoid glasscannon classes. Diablo 3 has a decent variety of builds that you can play to ensure every class is fun.
Alright you're level 1 and adventuring the depths of hardcore..what next?
Take it slowly, there's nobody around rushing you to move faster and have you trying to catch up to him/her.
You have the opportunity here to seek out those bonus events aside from the main quest. This helps prepare you for more challenging encounters by providing more xp and gold which you may use for skills and gear.
Stick together! I've seen so many people wander off and get destroyed. It is unfortunate but truth is if you're in a group of 4 and each go a separate way, you can't perform well[at any level] Mobs are harder to kill which results in slower xp/time and you can miss out on treasure goblins. Elites are not meant to be soloed by 1 person in a group of 4.
Same class as you in your group? Awesome--but try to use different skills, they won't stack. For example, if you have 2 monks go ahead and have 2 different mantras. If you have 2 barbs, get different shouts, etc. This ensures an increased benefit to the overall group.
Most common question I hear: "What if I die?"
Take it from any of us here on hardcore. Nobody wants to die and the game is played to avoid such a fate and most here believe the game is more enjoyable with the knowledge that death is permanent. If you die, simply brush yourself off, learn from your mistake, and remake your character. This provides an interesting dynamic in the game: Realism.
"What type of gear do I want?"
That depends on your class and skill build and whether you're going for more of a dps route or a support. The beauty of the game is that there is not only 1 way to play and this is the moment you will experiment with different items and skills to ultimately make yourself a successful hardcore player and have fun. Initially, I recommend stacking on vitality for all characters which will enable you to shrug off more blows and live longer. Great for inexperienced players with slower reaction times.
"Do I need to grind the same stuff?"
Not at all until you get to much harder difficulties of the game in which it may be useful to farm some materials to craft some new armor, or get lucky to find some epic lewtz, etc. If you're used to softcore, it should play the same way.
"What's the point of playing hardcore?"
Essentially the dynamic of realism mentioned above provides for a more thrilling game experience. A few D2 hardcore players will say "It's the only way to play diablo!"