According to Tobias, the first game played in 1870 was the Empire Club's anniversary game, held on April 24. Sadly, I couldn't find a contemporary reference to that game. However, the second game Tobias mentions was this match between the White Stockings of Chicago and the Unions of St. Louis. It was the first real match of the season and one heck of a way to start off the year.
The one thing I want to point out here is the price of admission, which was fifty cents. For some reason, I've always thought that admission to the Grand Avenue Grounds, during this era, was twenty-five cents but looking through my notes, I see all the advertisements I have state that admission was fifty cents. Now, it could just be that there was an increase in price when a big club came to town and all the advertising I have involves the bigger clubs that came to St. Louis in the late 1860s. It could just be confirmation bias. But, still, all the ads say fifty cents. That's something I'll have to keep an eye out for because, as I said, I always thought it was twenty-five cents to get into the Grand Avenue Grounds.