The Pokémon anime has had one series corresponding with each game generation, except for the first two (Red & Blue, Gold & Silver) which are classified as their own series. As you noted, Ash made the decision after the first series to start over with just Pikachu (and Aipom in Diamond & Pearl). However, they are intended to be one continuous story.
The most obvious example of this is whenever Ash returns to Pallet Town. Tracy Sketchit, a companion for a portion of the original series, reappeared in subsequent series as Professor Oak's lab assistant. Also at the lab are the majority of Ash's Pokémon that he caught in previous generations. Ash revisted several of them in the Best Wishes* episode 142, "The Dream Continues", and the Japanese version had a post-credits image of all Ash's Pokémon that were left with Professor Oak from all the series up to that point.
There are certainly other references to Ash's story being continuous. He'll occasionally run into characters that he's met before or reference previous events. There has never had a "reboot" of this story, even though each series largely repeats the same story arc.
So Ash is still the 10 year old** taking one journey after another into each region as he was in episode 1.
* The Best Wishes series corresponds to Pokémon Black & White.
** Yup, Ash doesn't age. But then again, neither does Bart Simpson.