Tel Aviv a good base to see Israel & Palestine?

I live in the Tel Aviv area, and I can confirm that most of Israel is easily accessible from there. I have no idea what tourist infrastructure exists in the Palestinian Authority area, if any, and how to get there from Tel Aviv.

Tel Aviv is something like Miami, in that attractions include the beach and a lively nightlife, while Jerusalem has a lot more historic attractions. Haifa, my hometown, is a beautiful city about 60 miles north of TLV, that overlooks a gorgeous bay and has the Bahai hanging gardens - definitely worth a visit! If you have 3 weeks, my suggestion is to work your way through Israel, sleeping a few nights in each region, don't just stay in Tel Aviv.

No. Stay in Tel-Aviv to visit Tel-Aviv and then move on. Tel-Aviv is a cool city, the coolest in Israel, but exploring the whole country from there is cumbersome. One of the highlights in Israel is Jersualem. Visiting Jerusalem takes some days. This means that you have to travel several times back and forth along the same road. The roads are congested. You will lose time in traffic jams. And the Israelis are aggressive drivers. So driving around is not really fun, especially if there is a lot of traffic. Now I have been talking about Jerusalem. But there are also sights further away, as the Dead Sea, Upper Galilee, the Golan heights, ...

I recommend that you make a plan of what you want to see, and then choose several bases accordingly. Choosing Tel-Aviv as a base to visit the whole country, plus parts of the West Bank does not make really sense.

Tel Aviv and Jerusalem are both large cities near the center of the country, which is to say that they are both good choices for you.

Tel Aviv has the advantage of being on the coast, and therefore closer to Haifa, the other large city in the north (away from the Gaza Strip), meaning that it is the better base for visiting Israel alone.

But Jerusalem is closer to Palestine (the non-Gaza part, that is), and is more historic. Based on my "read" of your interests, that could be a slightly better choice for these reason. Tel Aviv would be a close second, in any event, since they aren't that far apart, with the non-central Haifa, a distant third.

Yes, it have links to all major cities and regions by rail or by road

I think I would pick Jerusalem, but both would be decent choices, they are a short bus trip apart anyway.

It's not far but there are no good connections from Tel Aviv to Palestinian cities in the West Bank. If you rent a car it can be doable but check car rental conditions about going into Area A.

If you want to travel in the West Bank Jerusalem is by far the best place to be based. You can easily find a place to stay from where you can walk to both the Palestinian bus station in East Jerusalem and the Israeli bus station in West Jerusalem. For West Bank places this will allow you to leave early and return in the evening.

That said, as a city I personally much more enjoy Tel Aviv than Jerusalem. There's a beach and there's a much more developed nightlife. If you have three weeks it probably makes sense to spend a week based in Jerusalem (to travel in the West Bank) and one week in Tel Aviv. And then keep one week flexible to stay in whichever place you enjoyed most.