First time safari? Two lodges, 3 days plus 3 days would be right, being sure to pick very different types of scenery. You are out 8 hours a day at many places, so even that may be too much. Walking safaris are a bad choice for a first timer - you walk slow, animals disperse fast: you will see much more if you drive with a great guide.

Safaris are early AM and dusk affairs; many animals are thin on the ground during the hot part of the day. Since you haven't been on safari before, though, the "common" animals like elephant, zebra and giraffe will still be easy to see, likely even from your room if near a watering hole.

Safari camps are very expensive, especially in Botswana. Honestly, the highest luxury may not be required for a first timer. I have had camps ranging from public parks (in S Africa and Malawi) at $60 a day (crowded but can self drive), a private self-catering reserve in Botswana (at $150/pp/day, a fantastic value as we basically had the reserve to ourselves) to a higher end place in Zambia (at $400/pp/day) and up from there. The Singita lodges can easily get to 2k/pp/day or even higher.

My recommendation for first timers is Zambia: lots of animals (though rhinos are very thin on the ground), very cheap flights between parks on ProFlight (don't ignore this - it can be very expensive to get to some safari sites!), perfect shoulder weather, plus Vic Falls as a nightcap.