How and how much does it cost to I register my startup logo (International)

This may sound as a duplicate of other questions, but the enphasis is International. I've found it neccesary to have a logo at the stage of looking for VC's, among other things. But currently I do not know in which country I will be stablishing legally

My concern is that at this point I have (and it is likely for other entrepreneurs as well) participated in contests/incubators/events in different countries where the logo is inevitably exposed. So then maybe I should consider protecting it to avoid future inconvenients.

How can I register the logo so it is valid internationally? With this I mean so I have rights when I publish it on the Internet. And I would like to know an approximate of how much that would cost.

Trademarks need to be explicitly registered in each country to be eligible for protection. The cost is dependent on how many countries (and categories) you want to market your services and have the legal recourse against dilution. And that doesn't include policing it either.

participated in contests/incubators/events in different countries where the logo is inevitably exposed

perhaps it is a misappreciation of what is protected. You want you name/logo/tagline to be seen, but more importantly associated with your brand. Trademark law aims to prevent unfair practices such as passing off (pretending to be someone else) or deceptive conduct. So if you see someone using your registered trademark (r) without permission, you have legal recourse but it main outcome is to preserve some residual value in your marketing spend. Just registering it doesn't wave a magic wand to prevent someone else persuing the same marketspace under a different name or business model.