and Welcome to the forum!
Originally Posted by Michael.M
I'd say remove the title of "modern muslim." There is no such thing. You don't have to title yourself, but just think of yourself as a muslim who needs to improve in his worship just like everyone else. No one is perfect and we all have our shortcomings when it comes to worship. You are a Muslim who still needs to learn the basics of Islam. Many of us grew up Muslim, but our families were not religious, so we had to go through the same thing you're going through.
Also saying adhkar such as mashallah alhemdulillah etc. is a part of our nature, so don't think of it as "it's not in my nature." If someone gives you a gift, do you just take it and walk away? No, you'd say "Thank you so much, you're so kind, etc." So when you say these Islamic phrases you are also Thanking and glorifying God for everything you have and everything around you; and instead of saying "hi" which really has no meaning, you'd say assalamu alaikum because you're wishing your brothers and sisters peace in their lives. So do you not prefer something like that to be more of your nature?
My advice to you is to start incorporating what you already do know on a daily basis. Since you know muslims say things like "inshallah, mashallah, assalamu alaikum" then start using them because you receive good deeds from saying these things and they are considered as a type of worship. You will get closer to Allah subhanahu wa ta'ala the more you begin to acknowledge who you are through your actions. Learn how to pray as soon as possible because that is obligatory for you to do as a Muslim and it's the best way to connect to Allah and build reliance in Him.
As for being in the mosque, as a man you're supposed to pray the obligatory prayers there. In prayer, everyone has to have a focus so that they feel that "thing" you speak about (it's called khushoo'), no matter where they are at or who they're praying with. You can pray nafl and sunnah at home though after you pray with the congregation at the mosque. Praying congregation with others builds brotherhood and unity.
Don't think negatively of yourself just because you have not started practicing like you'd like to. You will get there inshallah, just put in more of the effort and make little changes everyday. Growing up, I didn't used to say adhkaar as often either, until I had a friend who would say them ALL the time. It started to become a habit of mine too, and then my habit rubbed on to others as well..so imagine the rewards of my friend now who was a constant reminder for me and now for all those other people too! :) You can be like that soon enough.