Tanzania is a wonderful choice.
Issue though... it's Christmas/New Year's where rates are the highest, busiest period and then availability. Whether on the mainland for safari or Zanzibar for the R&R.

Major City - in Tanzania? Well there is Dar-es-Salaam/DAR at southern end of Tanzania, but it's rather hectic, lots of traffic, though a few museums. Otherwise, there's Arusha town, about 1/hr west of Kilimanjaro/JRO the arrival airport and from where safaris commence. Small town, restaurants and cultural activities can be arranged.

With 10-14/days:
Day 1 - Arv. Kilimanjaro; transfer to Arusha - o/n
Day 2 - Arusha - Arusha Natl Park or village/orphanage visit - o/n
Day 3 - Drive to Tarangire, known for elephants - o/n
Day 4 - Tarangire - o/n
Day 5 - Drive to Ngorongoro Crater; afternoon crater tour - o/n
Day 6 - Drive via Oldupai Gorge to either Ndutu or Southern Serengeti, here to catch up with the migrating wildies - o/n
Days 7 & 8 - Ndutu or Southern Serengeti - o/n
Day 9 - Fly via Arusha; connect to flight to ZNZ - o/n
Days 10, 11 & 12 - ZNZ - at leisure - o/n
Day 13 - Transfer to ZNZ airport for flight to DAR or JRO for homebound.

Suggestion only, you can add another day in Serengeti/Ndutu and shorten ZNZ by one day... your decision.

From the States:
KLM flies into/out of JRO and DAR;
Swiss flies into Nairobi/NBO*.. out of DAR
Emirates flies into NBO*... out of DAR
Ethiopian** flies into JRO (via Addis Ababa)... out of ZNZ and/or DAR

From London:
British (BA) flies into/out of DAR; not a daily flight though; if arriving here, you can consider instead of the Northern Circuit itinerary I show, visit the southern parks as Selous and Ruaha... short flights from DAR, spending 3-4/days at each, then off to ZNZ. Otherwise, you'd need a short-haul internal flight from DAR to JRO/ARK to commence safari in north.
Virgin flies in/out of NBO
KLM flies from AMSterdam, where many choose a short flight from LHR to AMS for flight to JRO.

*more carriers into NBO and usually less expensive; arriving here, a connecting flight to JRO can be arranged; or an o/n in NBO and then flight next morning to JRO.
** Ethiopian has more competitive prices than KLM

Check Kayak www.kayak.com or www.kayak.co.uk for flight schedules/fares.

Lots of options, but best you get working on this quickly.

Consider contacting Nancy at:
Aim4Africa, www.aim4africa.com... she's UK based, but is Tanzanian and knows country as the back of her hand!

Good luck!

Many flights to Nairobi/Tanzanian now full. If you want to go..book very soon.