In reply to bencollins:
Firstly, because you say summer, you should be looking at the Karakorum, Hindu Kush, Tien Shan, parts of Northern India and parts of Tibet. The climbing season for these places is June to Sept (July, Aug best) Nepal will be washed out by the monsoon, so no point looking at the Nepalese 'trekking peaks'.
I've been on a mountaineering expedition to Tibet, and would not recommend it as a first Himalayan destination. Unless you have a lot of money and are willing to spend a lot of time planning and organising logistics. Some of the objectives out there would be great for you, but the circumstances lead to Tibet being poor value for money compared to other Himalayan destinations.
The Central Tien Shan (near Khan Tengri Base Camp) has one possibility. Peak Chapeav (spl?) about 6200m. This was my first big peak. It can be climbed from either the South or North Base Camps of Khan Tengri. It would be worth looking at the summitpost.org page for Khan Tengri. This peak could be climbed in just over 3 weeks from the UK, as a helicopter drops and picks up from BC. There are no permits (only one very cheap boarder zone permit less than £50) and the costs can be low (no permit, no porters, no cook, no guide, no high altitude porters etc) You just need to handle the logistics and food. ITMC is a good local agent in Kyrgyzstan. They quote about £600pp for the transfers from airport in Bishkek to BC and back to airport (inc. heli, trucks, buses, hotels etc)
Northern India, I know very little about, but I've heard that some of the peaks around Leh are good for a first 6000m peak. One example would be Stok Kangri.
Pakistan I think presents the best option for you. Some areas are 'open zone' meaning that you do not need a liaison officer (an expense and a pain) or a guide AND peaks below 6500m do not need a permit. So a peak below 6500m in an open zone can be very very cheap to climb. An area that I would recommend is Shimshal. Three peaks there spring to mind; Sonia Peak, Mingling Sar and Yazghil Sar. All are 6000m give or take, all are a nice trek from the village of Shimshal (which can be reached by road), all are technically easy (Sonia PD, Mingling Sar F/PD, Yazghil Sar AD). Cost can be very low and the mountains very accessible.
Fly to Islamabad, bus to Gilgit, bus/jeep to Shimshal can be done in a 2 to 4 days. Walk in to BC's range from 2 to 6 days and you would need a few porters to get gear/food to BC. Porters can be organised very easily at very short notice (ie turn up in the village one night, next morning 5 porters willing to go to BC). Food can be purchased in Islamabad or Gilgit at low prices. Transport is dirt cheap. Porters costs are fixed and are quite inexpensive. From the village of shimshal you are free to trek and climb (below 6500m) pretty much any ware.
Have a look at the summitpost.org pages for Sonia Peak and Yazghil Sar. I was in Shimshal this summer (independent, no guide, very few porters, low budget. It was me and 4 people new to climbing in the Greater Ranges, all summited Yasghil Sar, so I would think great for a first 6000ish peak), and Big Lee (posts on hear and maintains the Yazghil Sar page on summitpost) has been a couple of times (again low budget, no guide etc).
Also worth saying that travel in Pakistan is great, I have spent over 3 months out there on two expeditions and have never had any problems (security, tensions, being western etc)
Hope this helps and feel free to ask for more details etc.
