I've already said I recommend activated carbon, and not RO. What do I do personally? Water from the fridge filter, or straight from the tap (at current location they taste the same to me, and pretty good too)
I don't like RO because RO strips too many, basically all, dissolved substances and ions and everything from the water. Water with nothing in it is chemically out of balance with every environment, and it really wants ions back. It will corrode high grade stainless steel pipes in a few years in its desperation to get back ions. Your body doesn't need just water, it needs water with a very specific balance of minerals. Suffice to say RO water would be very detrimental to that balance. There are three ways to get around this 1) not really use RO but imply it's RO 2) reintroduce a portion of the RO reject stream to the treated water (hey 90% treatment is better than none) 3) introduce a mineral pack that must be regularly replenished. For a good system you get 3). But your body is biologically compatible with minerals in water from springs, streams, and rivers, why assume that the random engineered mineral pack is better than the naturally occurring minerals? Sort of like never exposing yourself to bacteria is counter productive and unnatural, only drinking artificial water seems short sighted. Plus you are taking something out just to pay to put something else of unknown quality back in on a regular basis.
Next RO wastes a lot of water, as previously. Plus, the RO membrane is very easy to damage or foul, and will need other filters in front, some of which could be substituted by industrial chemicals but not in a household RO. Even then, it will foul pretty fast and if you can't chemically clean it (household filters I assume are not designed to) then you have to throw it and the other filters away regularly despite being in otherwise good condition, just plugged. Very wasteful and expensive to maintain, and if not maintained could even develop biological growth and make things worse.
GAC is also somewhat wasteful and expensive, but not nearly so much. It is good for tertiary (taste/odor) treatment of many compounds. While YMMV may very depending on water chemistry, temperature, media, etc. (I'm not stamping this advice) it also can serve as primary treatment for a pretty impressive range of things including VOCs, DBPs, and even heavy metals. It's used to remove gold from cyanide (and gold is really attracted to cyanide), so I assume it catches gold silver and copper. It catches mercury, so I assume cadmium as well, and I assume some others. None of these should be significant in your water, but it won't hurt if an insignificant amount is significantly removed. Importantly though vs RO, it also leaves all the good ions intact. So default to doing nothing, but if your water tastes bad then GAC.
Note, I'm a nuts and bolts engineer not a chemist or anything like it. I've just worked on a number of water treatment projects.