You've come to the right place and the information you've received is accurate.
I would follow James's advice: get the travel card at the airport, then get the taxi. There is no need to worry about taking a taxi to your flat. They will assist you and you'll be in the heart of Pest in no time.
Regarding the ticket for the tram: this is not like the NYC subway system where you walk through a turnstile, there are no barriers to entry. You are on the honor system with ticketing.
James recommended the TravelCard which is inexpensive and allows access to all busses, trams, and metros (subways) in the city. A 24-hour travel card is about $5. You may be asked to show your travel card, so make sure you have it available at all times.
You can also purcase single tickets and those are validated in a machine. You could also be asked to show that ticket.
I find the TraveCard easier to manage than tickets. I have a clear phone case and place my travel card on the back of my phone where it is visible through the case. My last visit was in September and there were more checkpoints - both entering and exiting - than on previous visits. A quick flash of the card and I was on my way.
One thing to note: it is not the most senic drive from the airport into the city - don't worry!