Renown just for doing ultra casual stuff (one LFR or Normal clear for legendary recipes, campaign, callings, HC dungeon when you have quests) takes 1-2 week. Full 40 Renown I remind, last Renown with player power is 30 something.
Legendary is only problem if it drops with specific world boss, so there is choice to wait 3 weeks for bis or make second one.
Relics drops often and you have specific sources. Also most important is to find right ones, power differences between ilvls are not big.
Overall I would say geting barely minimum to raid heroic takes ~1.5 week of active play. For sure you rolfstomp open world at this point.