Is Iran safe to visit?

Short Answer: This is an unanswerable question.

We cannot know what you mean by safe. Even with your descriptives in the body of the question, you are asking about something that is personal to you and your beliefs. No one else can answer this in a way that would be completely acceptable to you.

The primary reason for this, in my own opinion is:

I cannot find any place where your own government posts any travel advisories for Indian travelers at all. It's not that there are none for Iran (or any other place) at this time. It's that there appears to be no place for Travel Advisories themselves to be posted. (If someone else finds a place where the Government of India posts Travel Advisories, I will quickly delete this whole answer, as that would be an objective answer.)

So, lacking any official advisories, you would have to accept advisories issued by other governments, such as that posted by Gayot Fow, or anecdotal comments by 'regular travelers' posted as answers. If they have travelled in Iran, then de facto they survived. If someone came here and said I had no problems, but there were daily demonstrations in Tehran against Western Imperialism, well, yeah. OK. So what? There are daily demonstrations here in Washington, D.C. about everything from war to the economy. All in all, everywhere in the world, on a personal level is the greatest place in the world, with the nicest kindest people everyone has ever met.

Every city has crime. An Indian is less likely to have troubles than a white European, who would be less likely to have troubles than me. But, as I said at the beginning, your own government doesn't appear to care to post advisories, so all you will get is anecdotes, which in the end, aren't really very useful.

Depending upon your nationality, there may be problems. According to the UK Foreign and Commonwealth Office, travel advice is separated into two areas...

... advise against all travel to within 100km of the entire Iran/Afghanistan border; within 10km of the entire Iran/Iraq border; the province of Sistan-Baluchistan; and the area east of the line running from Bam to Jask, including Bam.

The FCO advise against all but essential travel to the rest of Iran.

British travellers to Iran face greater risks than nationals of many other countries due to high levels of suspicion about the UK and the UK government’s limited ability to assist in any difficulty.

There’s a risk that British nationals could be arbitrarily detained in Iran despite their complete innocence, as happened most recently in 2011. In such cases the FCO has serious concerns that the subsequent judicial process falls below international standards.

In recent months there has been an increase in the detention of dual nationals on allegations of involvement in political and human rights activities. Dual nationality is not recognised in Iran. The Iranian authorities will deny that the British government has any legitimate responsibility for British Iranians and are unlikely to grant any consular access.

Independent travellers, especially (but not exclusively) if going off the beaten track, face greater risk than those in tour groups or business visitors.

You can always get updated information from the FCO; much of it is useful regardless of your nationality, but some is applicable to Brits only.

Source: https://www.gov.uk/foreign-travel-advice

For an alternate view,

Bahman Zenhari, a marketing executive with Iranian tour operator Pasargad Tours, said Americans have been visiting Iran for years — but numbers surged after the more-moderate President Hassan Rouhani was elected last year and replaced hardliner Mahmoud Ahmadinejad. He claimed there has been a 350 percent increase for tour demands in Iran since the election.

Source: http://www.nbcnews.com/news/world/americans-check-out-irans-golden-eagle-danube-express-tourist-train-n246816