Berlin airport pickup and tours

Assuming you are arriving at TXL ( Tegel) and this is where you are going, --- https://www.ihg.com/intercontinental/hotels/us/en/berlin/berha/hoteldetail/directions -- it is a bit over 5 miles, ( 8.4 km)

Transportation choices depend on your budget and size of party

I haven't taken any tours in Berlin, but I'm sure others have. What type of tour are you interested in? That will help responders provide suitable suggestions.

Berlin has a lot of historical sites. Many of them have museum-quality explanatory material posted, so I'm not sure a tour would add a huge amount of value in many cases. Because of heavy wartime damage, much of the architecture in Berlin is fairly modern; if you take a walking tour, you won't necessarily be seeing a lot of beautiful old buildings. And of course Berlin is huge and terribly spread out, so any tour's coverage is going to be limited.

I imagine there are walking tours focusing on the WWII period and others on the Communist period. Those could be very interesting, but it would be possible to spend 3 or 4 days in Berlin just at the key 20th-century historical sites.

Perhaps a trip to Potsdam would suit you. I haven't been there, but I understand it's very pretty as well as being of historical importance.

A taxi might be the simplest way to the Intercontinental, and not very expensive. Taxifarefinder indicates about 25 EUR. For 2.80 EUR, you could take the X9 bus to Zoologischer Garten Bahnhof, and walk about 1/2 mile (10 minutes) to the Intercontinental, or change to the 200 bus and go 2 stops, which will drop you across the street from your hotel.

I am sure the hotel concierge will be happy to book any tours after you tell them your interests.

Take a taxi from the airport to your hotel.
There are tons of tour companies in Berlin. Everything from private tours to huge group tours including those "free" tours that aren't. It depends on how much you want to pay and your interests.
For something small, try Jeremy, the Berlin Expert. This is private but his prices are not out of line. If that is still too expensive, go with Insider Tours. They are the top rated walking tour in Berlin. Have gone on 5 of their tours and enjoyed all of them. You might have a look on Trip Advisor to see what else is available, but book directly with the company, not with Trip Advisor as this is really Viator. You will save the tour company from paying a 20-30% commission to Viator/Trip Advisor.

From a local: Get a taxi to hotel.

Alternative public transport: buses X9 and bus 200 (stop infront of hotel).

On your first day in Berlin I suggest a walk to VisitBerlin in Europa Center. After that a HoHo bus tour will start close, drives you around and give some basic explanations. Just enjoy to hop-off and dive in.

On day 2 a walking tour through the Eastern center makes sense and will give further insights. Checked myself Insider Walking Tour which presented 80-85% correctly. As tourist you will not mind the gaps / mistakes. If you will find a tour with people who live long enough in Berlin or are even born here I would choose them because they can transport a feeling how it was in younger history, e.g. opening of the wall or living in Berlin during Cold War.