There are multiple ways to promote a local business on Google Map.

1- List your local Business with “Google my business” . You can start here https://business.google.com/ . Just singup with your Gmail id and follow the instruction to register your business.
You need to add your
Business name
Address
Phone no
Service/product categories
and some other info.

Then Google will send you a postcard with a PIN to verify your business listing.

The easy way is use “Google my business” mobile app and try to register your business.

Once your business is approved, it will be visible in local listing in Google map. With better customer support, ratings etc, you can start ranking at better position and you will get more visitors, leads, inquiries.

2- Another way is to use Google ads. Singup for Google ads here – http://ads.google.com/ . You can create a campaigns ( search, smarts ads etc) to promote your local business. I will recommend you to have a website to run the Google ads.

Except Google, Local businesses can also be listed on other websites like Justdial, India mart, Facebook etc. Also make social page on all popular platforms.