I use to have the PUR filter that attaches to the faucet. It was good because it was practical for cooking, had an indicator light for changing, and never had to fill a pitcher.
I kind of remember there being lead specific filters and regular filters, but I could be wrong. It was clearly marked either way.
https://www.amazon.com/PUR-PFM400H-Chrome-Horizontal-MineralClear/dp/B01I3D6DZ6
I use this. Works pretty well for me. You have to have the right kind of faucet so you’d want to check on that.
PUR Plus Filtration System, Chrome – Horizontal Faucet Mount for Crisp, Refreshing Water, PFM400H https://www.amazon.com/dp/B01I3D6DZ6/ref=cm_sw_r_cp_api_i_A63M8HH8KX42KE9JCYEN
Theres that one and this ones slightly different
PUR PLUS Faucet Mount Water Filtration System, Stainless Steel – Vertical Faucet Mount for Crisp, Refreshing Water, FM4000B https://www.amazon.com/dp/B00G15AA30/ref=cm_sw_r_cp_api_i_441Q15WA67WDV8DV3BD9
If you don't have good tap water, invest in a PÜR filter that can be attached directly to the tap. In order to brew a great cup of tea, you must start with good quality water. The water in my apartment is very hard and had a terrible mineral taste to it, and so I made a small investment into a good water filter, and it makes much better tea.
The temperature of water that you use will also effect the outcome of your tea. Some teas, like green and white, need lower water temperatures than black and oolong teas. This is where a variable temperature kettle comes in handy. I bought mine on Amazon for only $35, and it can be programmed to brew to a specific water temperature.
As far as the teas go, I'm sure others have mentioned this, but using quality loose leaf tea is the way to go. Tea bags are filled with poor quality "tea dust", and won't taste very nice. Try purchasing some fresh loose leaf tea samplers from online vendors. Check out the vendor list or Marketing Monday thread we have on /r/tea for some ideas.
Lastly, check out Mei Leaf's brewing guide for an idea on how to brew your tea.
I like these Pur water filters that attach to the faucet: https://www.amazon.com/PUR-PFM400H-Horizontal-Filtration-Mineral/dp/B01I3D6DZ6
They're very convenient (easier to use than pitchers, and don't require any difficult/permanent installation work). I buy replacement filters at Costco.
I haven't noticed a Chlorine smell in my tap water (except after my roommate poured bleach down the sink). According to the map in the "Desal Water Quality and Distribution Area" pdf at https://www.santabarbaraca.gov/gov/depts/pw/resources/system/sources/desalination.asp my location gets 20-40% from desalination.