I recently got a six week old bunny and I want to know if this is the right age to tame her to wear a harness? And can I train her to not be afraid of public places by bringing her outdoors in a carrier?
I reckon this is your very first rabbit? Take it easy.
6 weeks is very young, weaning already is stress on her system, changing envirnment too. First she needs to settle in, get used to you and her new home, her new life away from her sinlings. And grow out of the toddler age.
You don't really "tame" rabbits. To some degree you can get them used to things, like being handled, but it very much depends on the character what is possible and what not - for example, my 8yo Red still refuses being petted. Respect when the rabbit has enough and wants to hop away.
The things you mention are out of the comfort zones of most rabbits, and not things rabbits usually enjoy - unlike some very cute, popular youtube videos suggest, quite some misconceptions about rabbits stem from those.
Getting them used to be in a pet carrier is something good to train - keeping it in her pen, putting a treat in there etc - , that will help when going to the vet. For toting around, well, not so much. Some rabbits might tolerate it to some degree, but it's not really a rabbit thing. Harness and leashes are another matter, and even if you got one of the rare rabbits that don't mind that it's not like walking a dog or even a cat.
What I would do now: First get to know each other. Patience. You'll need a lot of this. Spend time close tho her, in her pen, to get her used to your presence. Small treats (I use their pellets as treats) help to break the ice. Let her come to you, build trust. Petting and cuddling is ok, That can take weeks, months, and quite likely there are some bumps ahaed on the road when she hits puberty at about 10-16 weeks. Consider getting her spayed then, especially if she's an indoor bunny, start looking for a vet that suits you in time.
No point in getting ahaed of things now, but time flies. Now, just take it easy, get used to your new companion, spend time with her and don't stress her.
Not a good idea at all. They are not dogs. Leave her at home in a large area with lots of enrichment. There is no need for her to go out and most bunnies get terribly stressed when taken out of their familiar environment, they are not like dogs at all. Besides, harnesses are not recommended, they are dangerous and can cause horrific injuries.
As mentioned, rabbits are very different creatures from others. There really isn't such a thing as "early training" in that an individual rabbit's unique temperament is not going to be influenced in any significant way to... not be shy, or enjoy handling, or liking new places. It just isn't how rabbits are. If anything, too much early handling can cause a rabbit to become more skittish and fearful of humans.
Patience is key with a rabbit. They stress easily and 6 weeks is very, very young. Many states don't even allow the sale of rabbits before 8 weeks of age. So for now, just let her adjust to her new home.
The following link offers tips for bonding with your bunny: