Frankfurt is always thought of being a major financial hub in Germany and hence to be visited only for business and not tourism. Arguably, there is nothing ‘must see’ in Frankfurt itself, but given its location, Frankfurt can be the base for day trips to several interesting locales in Germany and in Europe. More importantly, Frankfurt is directly connected to most parts of the world which means a convenient direct flight.
How to get to the city centre from the airport
Frankfurt airport is the busiest airport in Germany and not too far from city centre(less than 15 kms). I must admit I had bit of a difficulty in figuring out how to get to the train station and hence opted for the easier alternative-engaged a taxi!
The train tickets(S8 and S9 lines) to main railway station – Frankfurt (Main) Hauptbahnhof costs less than 5 € per person one way and the journey takes ~15 minutes. There are trains that run only between airport and Frankfurt Hbf while many are commuter trains. From there, depending on where your hotel is, you can take walk or take a taxi or the tram or the bus. The taxi set me back by 55 €, and took about 30 minutes. You can also take the bus from the airport and the cost is less than 5€ per person. Overall, the taxi will be the quickest option.
By the time I spent a week in Frankfurt, I got quite knowledgeable about the train service and could comfortably take the train to the airport for the return journey!
Public transport
Frankfurt has an amazing and extensive network of public transport comprising Trams, Metro(U-Bahn), buses and commuter trains(S-Bahn, the suburban trains). Taxis are generally not necessary. If you plan to use the public transport, do ensure you spend a few minutes understanding the fare zones and different modes of transport. Especially note that S-Bahn and Regional trains(RB, RE) are different.

I primarily used the trams and metro for travel within the city, and each station normally contains a map and a ticket vending machine that allows for displays in English also. At one or two remote stops, I did not notice any ticket vending machine. But the best thing to do is to buy a day pass, which costs marginally more than two single journeys.
Details are all available online as well, for those who like planning their trips to perfection!
Accomodation
Frankfurt Hbf is the hub of public transport and hence makes sense to stay close by. Plenty of hotels available, to suit every budget, but all of them will not be walking distance. Just make sure the hotel is close to a tram or metro line if you don’t wish to spend on taxis.
3 star accommodation with breakfast can cost upto INR 10k per night, with 5 star hotels above INR 15 k per night. Actual rates will of course vary depending on various other factors.
Vegetarian food
Easily available. Several Indian restaurants besides Mexican and Italian, not to forget the ubiquitous Subway. If you traveling away to the suburbs or on a day trip, better to check out online. I did not as I am not too fussy about where I eat, so long as it is vegetarian. On my trip to Rhine valley, I had lunch in a lovely Italian joint overlooking the Rhine river but when I went to Bad Homburg, I found an Indian restaurant run by a German family! You will have a choice in Heidelberg also.
A simple vegetarian meal in a fast food kind of a joint can cost ~10 Euros or so and ~20 Euros and more in a decent restaurant for one person.
For beer lovers, Germany is probably one of the best countries to visit, and so also for wine aficionados.
What to do in Frankfurt
Frankfurt was almost fully destroyed during World War II and has been rebuilt since. But few areas do retain the old world charm.
Romerberg, with its colorful buildings. It is a public square and on Sundays, it is carnival atmosphere with several rides, games, eateries coming up here and in the path to and next to Main river. The Romer building, which is over 600 years old, is located here.
Frankfurt Cathedral, believed to be over 700 years old, and has been rebuilt atleast twice. Similarly, the Opera house too has been rebuilt over the years.

Palmengarten, the botanical park.
Stroll around Old Sachsenhausen or Hauptwache or along the bank of Main river. You can even walk across the Eiserner Steg or the Iron Bridge, spanning across Main river and famous for its love locks, thousands of locks clipped to the railings. Lot of people jog or cycle on the banks of the river and on Sundays, many take to boating.

Besides the above, there are Museums, zoo, Main Tower with a viewing gallery at 600m and Goethe House(after the most famous German poet). You can also take a cruise on river Main. And my favourite pastime is generally to take a tram and get off at places that look interesting. But overall, nothing much for extended stays.

There is a Hop on, Hop off bus connecting most of the tourist places but you can also get to all the locations by public transport.
Day trips
Frankfurt is ideally placed for several half or full day trips. The top ones to consider, in my view, include the following:
Rhine valley. The Rhine river is over 1,300 kms long but the most scenic section-the Upper Middel Rhine Valley- lies in Germany. Read more..
Black forest area, famous for thermal springs, cuckoo clocks and black forest cherry cake.
Rothenburg, a medieval town
Heidelberg, the old city famous for its castle.
Other smaller towns within an hour or so by public transport such as Bad Homburg, Darmstadt etc. There will be nothing great to do but the feel of strolling around in a small town is good, though I will readily admit these will not be on a regular tourist’s itinerary.