A mug and an infuser are the minimum, but it's good to purchase a small inexpensive gram scale. I brew my morning sencha with just an 8oz glass mug and a Finum infuser. Your water heating unit is good for a start. This setup will let you brew almost any tea you want to try.

regardless of mesh, and I have two very good ones I'll send link, one-person size, with shou lately I have been also pouring through just normal good grade coffee filter using a one-cup drip-coffee porcelain 'cone' that fits very well with a variety of cups/mugs. In fact, a few days before this post I had taken pictures of that rig to post here, pursuant to the discourse on fungi and bacteria in pu, but got lazy.. With my current rotation of shou pu, I rather like the clarified much better.

edits/adds: I own and have extensively used both of these and reco, except that the first one is now unavailable via Amazon https://www.amazon.com/gp/product/B01N1OTXHW

https://www.amazon.com/gp/product/B000I68NCS Medium Size These both do a very good job, and both have proven to be REAL, high-grade stainless steel over years of use and countless steeps. The Finum is my travel-screen as it is a tad shorter, and very light, and near-bombproof. The Finum is also a little easier to clean as the other has a seam on the upper band that is very smooth, but still will make dragging out the last bits a litte more tedious... like 2 seconds more. These are both completely 'inert' adding nothing to taste, subtracting nothing but what you want it to. I am concerned that anything finer than this will make precision of your steep temps and times trickier because, at some stage, you're going to strain, and if that stage takes seconds long to process, then it must be factored-in. These strike the right compromise to me

Edit 2: Don't let the structure of the Finum mislead one into thinking that the ribs on the side intrude: they do not. The interior is very smooth, very easy to cleanup, and polish occasionally with baking soda.