I recently finished all three seasons of Hajime no Ippo and I wanted to know where the last season left off in the manga. Also, I wanted to know if the anime follows the manga storyline or if there are things changed, so as to know if I have to start the manga from chapter 1 or not.
Rising ends with a four-episode arc focused on Kamogawa and Nekota in their younger years. Though the anime places this arc after Takamura's victory against David Eagle, it originally came after Takamura's first match against Bryan Hawk in the manga, so you'd want to pick up from the end of Takamura's fight against Eagle, which ends on Chapter/Fight 556 in the manga.
Most people seem to agree that while the anime made some cuts for the sake of time, important scenes have been retained, and that cut scenes generally would have added depth of background, humor, or added characterization that the series had no time for or that would have disturbed the flow of the plot. According to the first link, a fight between Hayami and Kobashi that originally took place in Chapter 290 of the manga was cut, but it was relatively short and likely had little bearing on the main plot.
I haven't actually watched it, so I can't say for certain, but from what I've read, the anime should be a pretty faithful adaptation of the manga. If you really wanted to, you could start from Chapter 1 anyway to pick up the minor details that were cut. You may end up enjoying the series more, or it might not make much of a difference.