If you just want to do current raiding
right now, you can buy on the Mogstation a level 70 boost (called
One Hero's Journey) for $25, and a story skip (called
Tales of Adventure: Stormblood) for $25, allowing you to skip all story content up to Shadowbringers. If you skip cutscenes, you can complete Shadowbringers in about 15-20 hours, and then you can start raiding current content (Eden savage difficulty, Ultima Ultimate, Bahamut Ultimate, and Alexander Ultimate).
If you don't want to buy boosts or story skips, you can
get to Shadowbringers in 60 hours if you skip cutscenes and don't do side content.
For raiding, the most demanded jobs are tanks and healers, ofcourse. Any job can solo the story content, but if you level only a healer, it will go slower because they deal a lot less damage than either DPS or tanks. Thus, if you're going to level a healer, I'd recommend you level a Scholar, as experience is shared with Summoner. You can easily switch to Summoner to do that pesky story content.