I need something to let people know that my business exists. So I've been thinking of printing pamphlets and distributing them. Is that a good start?
That is a GREAT start. Flyers work well with a cleaning business, even today with Google and everything. Keep working the streets in your demographic and focus on talking with people and getting to know them on a personal level.
Why would someone look at Google when they have your flyer in their hand?
However, having a website is good too (landing page is better with just phone number and some basic information about your business), but it is very hard to get to the top of Google search and hard to stay there if you do.
Instead of spending months and months or even years on SEO
(Search Engine Optimization) to get to the top of Google, verify your business with Google Maps so that you show up when someone searches " cleaning company near me". It's easy and free from Google https://www.google.com/business/
You must have an address to do this, but you can use your home address until you get office space (if you ever do, I never did).
You also may want to add your business to websites like Angie's List and Yelp.
Oh, and yes order business cards. The simpler, the better.
DO NOT PUT YOUR WEBSITE ON YOUR BUSINESS CARD. You want people to call you instead of going online to find hundreds of other cleaning companies.