Pretty Guardian Sailor Moon is currently airing on Japanese TV at 7.30am every Saturday morning. If you live in Japan, you can go to this page to see what station it's on in your area.
However, if you're reading an English fan site, chances are you don't live in Japan. Luckily, it's possible to download the episodes quite soon after they're aired in Japan. For those of us who can't speak Japanese, a synopsis is posted online at Genvid.com soon afterwards as well, and so far fansubs of the episodes (versions subtitled by fans) have been made available within a week or two of each episode's airing.
Raw Episodes:
Within 24 hours of each episode airing, this page at Saiyaman.net has a torrent link to download the episode. To download files from this site you will need BitTorrent. The BitTorrent site provides detailed help for using the program, but basically, you click the torrent link on the site, and the program downloads the file from "seeds" (people who have already downloaded the entire file). Anyway, the quality of the Saiyaman raws isn't brilliant, but they're the quickest way to get new episodes.
Higher-quality versions of the raw episodes (which aren't available as quickly), as well as raw versions of older episodes, can be downloading using IRC. If you use Windows, chances are the IRC program you want to download is mIRC. If you need help using IRC, you should be able to find the information you need either on the mIRC site, or on this site. The IRC channel I would recommend downloading from is #pgsm on irc.mircx.com. There are lots of fserves with lots of PGSM downloads.
Also, if you are willing to fork out large amounts of money, you can buy the DVDs that are being released in Japan starting from March. If you live outside Japan, websites you can buy them from include CD Japan, YesAsia and Amazon.co.jp.
Fansubbed Episodes:
Pretty Guardian Sailor Moon is being fansubbed by TV-Nihon, who by most accounts are doing a pretty good job. Recently they have been making fansubbed versions of each episode available within a week of them airing on TV. The BitTorrent links for TV-Nihon's PGSM fansubs can be found at http://bt.base0.net/tv-n/. Incidentally, TV-Nihon has also subbed the "Making of" special that was shown around the time the series started.
There was also another group which fansubbed the first three episodes. The people involved didn't like the show very much, and randomly replaced lines with references to things like tentacle hentai (if you need to ask what that is, you probably don't want to know). In my opinion the jokes during the episodes themselves weren't all that funny, but the joke subtitles for the commercials in Act 1 were hilarious. You may see this fansubbed version referred to as the "partial parody" version. It has been commented that this version actually has a more accurate translation than TV-Nihon's subtitles in most cases, but not having a very good knowledge of Japanese, I can't confirm this. This version wasn't officially by one particular group, but it's the Anime-Otakus staff that worked on it. The BitTorrent links to download the files could previously be found at http://a.scarywater.net/ao/, but they seem to be gone now.
There are some other groups that are fansubbing PGSM, but as of yet no one is as up-to-date as TV-Nihon. If you wish to contribute details of other fansubs and how they can be acquired, feel free to e-mail me so I can make this page more comprehensive.