I can't answer you re the camel festival I'm afraid. All I would say about Pushkar is that it is a real tourist town, the most touristy I came across in India apart from places in Goa - which is a shame as it's a beautiful town and is a bit ruined by the tourism. So I would maybe follow that advice and plan to arrive early on, once you've got the dates clear.
I don't think you can really camp in Pushkar (maybe I'm wrong but I didn't see any campers there) but budget accommodation is very cheap there - I would look in Lonely Planet for the recommended ones and book ahead if it'll be at the time of the camel festival, because places will be busy - Lonely Planet will have phone numbers and if they provide phone numbers, the person who answers will have enough English to understand you.
Re wanting to see the 'real' India. I'd say as long as you stay away from most of Goa (and I would have said Pushkar, except you want to go there!), you WILL see the real India. Cities, towns, temples and sights (maybe apart from the Taj Mahal - and Pushkar!) don't get drowned by tourism in the way they do in other countries - even if lots of tourists flock to these places, India itself is everywhere - you will always be vastly outnumbered by Indian natives, and just spending days walking around is enough to get a real experience. I didn't actually visit that many temples while I was in India (and I was there for 6 months) - I saw a few, but generally it was enough for me just to be in a town, experiencing it in other ways - eating, taking buses, walking around, watching people and what's going on around - even going to the post office was an adventure.
Re getting to Pushkar from Delhi: You'll need to get a train from Delhi to Jaipur, then take a bus from Jaipur to Pushkar (which stops at Ajmer, where you may or may not need to change buses, depends which one you go for). It's only a few hours between Jaipur and Pushkar.
After Pushkar - well, I would recommend Rajasthan. Although it's a well worn tourist route, I thought it was beautiful and fascinating. In terms of sights, yes, every major place will be plugging their forts, but, well, the forts are good to see! But so are the towns themselves. I had a great time in Jodhpur, and wouldn't have felt like I'd missed anything if I hadn't seen the fort in the end. The city itself was interesting enough, with a big market square, and nice walks around the city and among the alleys of the blue buildings. Udaipur is also worth a visit. Jaisalmer is touristy again, though not half as bad as Pushkar, but attracts tourists because of camel safaris. Bikaner is worth a visit. I think you could stop at 2 or 3 chosen places in Rajasthan in a week to 10 days. To be honest, if you only have 3 weeks, I wouldn't even take flights south - there are amazing places to see in south India, but in such short time, you won't appreciate them - save it for your next trip there. You can get from one place to the next in Rajasthan on buses which take 5 or 6 hours between places. So it's not too bad.
After Rajasthan you could head up to Amritsar in the Punjab (you can get a bus or train from Bikaner). Yes yes home of the Golden Temple and all - but it is well worth seeing. I loved the city and I found it fascinating at the temple (I never actually went inside). There are western tourists there, but mainly Sikh pilgrims - just walk around it and soak up the atmosphere. Like i said, I never even went in the temple, and I had a really full experience there, talking to the pilgrims, just wandering around, watching the rituals etc. You can stay in a special 'foreigners' dorm in the temple complex there, for up to 3 nights, and they only ask for a donation - you even get your meals - so if you're short of money, that's a good option.
From Amritsar you can take a bus up to Dharamsala (about 6-7 hours) and spend a few days there. Ok - a bit tourist travelled again, but worth it, I think.
And end with a few days in Delhi before you go back - Delhi's a better experience at the end of your trip rather than the start as you'll find it very full on.
Hope that's helpful, unless you REALLY want to get right off the beaten track - difficult to know. But just to stress again - apart from travellers enclaves, you'll find tourism different in India (there's still heavy touting of course). India manages to stay being India, despite the tourism - you'll see. 😊
PS re travelling alone - if you can take more time, really DO! I travelled India alone for 6 months and was absolutely fine - most of the time I ended up hooking up with other travellers, and when I didn't, I felt quite safe, and managed to find my way round all right. So think about taking longer, then you can really explore the place. India is very set up for travellers so you won't find it anywhere near as challenging as what's going on in your mind, honestly - I actually found it pretty easy, and you will too. If you don't know if you'll ever manage to get back there, then make the most of going there when you do - you won't regret it
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