Trade Mark and other protection

At the time of writing this i have not had any quotes and have not set a budge as I will just be going with the best offer so please ignore the budgePpit is only there as PPH forces me to set on.

I have a name for a software web and phone application that I am having developed. I have registered the website domain I want to use. I want to protect the name of my app. There is no Trademark yet but one could easily be created if that was a way to strengthen the protection of the name. there are 2 versions of the name I want to protect the name itself and then name prefixed with "my" so if the name I wanted to protect [which obviously I'm not going to reveal here] was for example say; garden I also want to protect MyGarden as a version of the name.

I am in the UK and would like to protect it in the UK for certain and likely in the rest of the world / english speaking countries depending on cost etc

I am likely to be able to do this myself. i have check n the gov website and the name spelt as I have spelt it is not used at all. similar sounding names are the for the short version of the name I want to protect but when I add the "my" to it there are zero results of a match.

I think it is only the name/s that I need to protect for the classifications that I will use them for but if there is anything else please mention.