You may have created a new Guardian and reached the Tower but Zavala, Drifter and Shaxx wouldn't immediately offer you their associated bounties. Bungie's clarification is not exactly great on this but it's not hard to get the bounties to pop up.
Many quests in Destiny 2 a shared between characters but some are not. There is no clear pointers that would help players distinguish these quests from one another but this guide might help in at least one portion.
For example, the Introduction quests are mandatory on each new character and you have to walk between the NPCs and complete these quests each time you have a new Guardian. Otherwise, the markers will just remain there to pester you. The good news is that you can pretty much skip any quests if you just want to do bounties.
Regardless of the other quest progress on your character, queuing up for a Vanguard Strike playlist or Nightfall and completing one Strike will prompt you to talk to Zavala afterwards. He will give you a quest completion reward and then offer three introductory bounties. Take one or all of them, get into another Strike and complete at least one bounty. Upon visiting Zavala the next time, he will offer the regular daily and weekly bounties.
It's essentially the same process with Shaxx and Drifter. You queue up for Crucible or Gambit, complete it, go back to the Tower, grab bounties and so on. The main annoyance, in this case, is that you will have to go through many loading screens, which is why I would suggest installing or moving Destiny 2 to an SSD.
If you are unsure why you would want to have alts on your account instead of focusing on the Guardian you love and enjoy, one of the reasons is that you might be short on Bright Dust. Destiny 2 veterans likely already know these mechanics and have a full stack of Bright Dust lying around but New Light players can farm up to 1,200 Bright Dust per character each week.
Keep in mind that if you complete weekly bounties on one class, you will not be able to pick them up again on the same class until the next weekly reset. For example, if you have two Titans and complete weeklies on one, the other one will not be able to pick them up. On the other hand, you can first grab all weekly bounties on both characters and then complete them for maximum Bright Dust gains.
Having three characters of different classes will result in you not having these issues.