Best ways to send suggestions and feedback to big branded companies

Well, in my view and as far as my own experience goes- going public with your points would be the best option that works. Twitter is one such better medium of connecting with bigger brands. Though you have the restrictions of having to bring up your opinion in just 140 characters, I still prefer it as a better alternative to emails and feedback links on their respective websites.
When I say Twitter, please note that I do not mean the Direct Message feature on the social media. Rather I would ask you go with a normal tweet on the official twitter handle of the concerned firm. Direct messages would not go public. Going public makes your feedback know more important. Do not hesitate to use stronger words if it is some sort of a complaint. You can always be straightforward without being aggressive and/or abusive.

You can also make use of the feedback option on the website if one exists. If they are really taking the feedback seriously, you can be sure of your concerns addressed.

Most of the popular brands these days have social media profiles. Let's assume you wish to contact Nestle group for their specific product. In such case you have contact email for consumer complaints and feedback. Other than traditional email you can access social profiles like Twitter, Facebook and Instagram.

Brands are more active on twitter and Facebook. Sending them direct message on Twitter often gets your suggestion or complaints to the quality team. However do note that social media profile are operated by third party people. So there is no assurance of letting your complaint or feedback reach to the brand.

Facebook pages if seriously operated by the brand then you should definitely try messaging the page owners.

My personal experience with brands like Hostgator, Evernote, and few startup companies was good. So I'd say try social media if you don't get complaint to reach to concerned department.