Judging from your post (which contains an English equivalent of the shahada), it's possible you're already a Muslim. However, it's expected to say it in Arabic.
The process is described here: How would I convert to Islam and become a Muslim?
There's not really a "correct" way: the important part is converting to Islam, not the environment it's done in. The recommendations are:
To have a good understanding of Islam.
While it's not required, it's preferred to convert (say the shahada) in front of witnesses. (Many Muslims think human witnesses are required, and may question you down the line.) It may also help to obtain an "Islamic certificate" to demonstrate you're Muslim when later going on hajj.
Converting in a mosque is normal: the imam knows the exact procedure and can help you through it, and you can make some friends.
Online resources about this are: JustAskIslam and NewMuslims.com.
The main difficulty is becoming a good Muslim: a convert needs to learn how to pray. Ramadan starts in slightly over a month, where there's fasting and zakat. Converts often begin with much "zeal of the newly converted", but it's a challenge to stay strong over a long period of time.
There are YouTube videos showing people converting to Islam in mosques, such as:
Australian embrace islam and take shahada in the mosque
LONDON GIRL CONVERT TO ISLAM IN ALBANIAN MOSQUE