How do I get a 1960s film look, like technicolor?
Pretty much any professional grade video editing software will have "color Correction/Color grading" capabilities -- this is the part of the art that you are looking for. Obviously, the "grain" you'll get here is digital/simulated noise, and not actual particles of chemicals. this is the Instagram filter before there was Instagram. 

the website "shotonwhat?" is a good resource to see which actual cameras (and lenses, but not always) were used for movies. Lenses, lighting, makeup and wardrobe and production design also make a big impact on the look of the film, some of which you've mentioned.

Another big part of the equation is the "film stock" -- the actual film used in 35mm/16mm/8mm cameras -- which had countless varieties based on different factors. Individual film stocks often had special characteristics (sometimes intended, sometimes not) which were favored by cinematographers/directors. 

the digital equivalent of a film stock is called a LUT (a look up table). so the way it works is this : you shoot the scene in the digital camera's "flat" mode to get the most information on your chip. Then you apply the LUT to make it look like how you want to. On set, you can live preview your LUT to approximate how it's going to look like in post, unless you're just playing with it in post from scratch. 

It's hard to give a specific answer to your question without spraying a bunch of brand names -- which is frowned upon here -- but I hope my answer helps you in farther research.