Yes and yes, although a bit later. Around 1863, when the first Laws of the Game of association football were codified in England, there were several competing sets of rules in existence. Some of these different codes had set up their own competitions before the first football (in the modern sense) tournament took place, but generally speaking, in addition to not being soccer competitions proper, they were also fairly minor affairs such as annual games between rival universities.

The world's oldest football competition, The Football Association Challenge Cup, was established in the 1871-72 season. Although you might be better acquainted with the more common name that it has kept to this day – the FA Cup. Wanderers FC beat Royal Engineers 1:0 to become the winners of the knock out tournament. Until then, clubs met on a more or less irregular basis with no set organized competition using the association football set of rules.

As for international football competitions, those would come after several years. Individual games between England and Scotland were held even before the founding of the FA Cup, as the two countries met in a series of five games between 1870 and 1872, but these are not considered to be official international games by FIFA as the Scottish side was represented by players based in England, and the rest of their country did not yet fully embrace the association football rules. It was only in October 1872 when the first officially recognized international match took place in Glasgow, a goalless draw between Scotland and England.

The first international tournament took place in 1884, twenty one years after what we consider to be the birth of modern football. It was called the British Home Championship and as the name suggests, England, Scotland, Wales and Ireland took part in it. The competition was born out of a meeting of these nations' associations in 1882, where they agreed on a common set of football rules. While games between their nationals teams had been played before, they all used their own slight variations until then.

So yes, in a few short years after the Laws of the Game had been published, clubs started to compete in the FA Cup, and then a while later, the first international tournament was organized. Check out the book The Ball is Round for for a more in depth look, or even the FIFA website for a good overview of the history of football.