With the release of the nineteenth game last week, known as Unfinished Dream of All Living Ghost, I'd like to revive this thread and help people get into the franchise.
If you would like to start playing the games, you have a few good choices. The first is to play the first game for Windows, known as The Embodiment of Scarlet Devil; this is when the franchise took a soft reboot after the PC-98 games and it features some of the most recognizable characters in the whole series, including the maid Izayoi Sakuya and the ice fairy Cirno. It's about an average difficulty of a Touhou game, but it was meant to be an introduction for Windows players and a new beginning so it's generally considered to be not too bad. Another option is Imperishable Night, the 8th game in the series and considered one of the easiest to beat. For a personal favorite I've started my journey with the tenth entry, Mountain of Faith, which has turned out to be a good starting point as well as being the first entry available on Steam. Of course, you can find all the games for free on moriyashrine.org as well as english patches on touhouwiki.net for ease of use.
There is a lot more to the Touhou Project besides the games, of course. There are several official manga series you can read; I have been recommended Touhou Suzunaan ~ Forbidden Scrollery as well as Touhou Ibara Kasen ~ Wild and Horned Hermit. Both are considered to be a good place to start for newcomers, as others assume you know the events of specific games.
If you have engaged with particular media and enjoyed, don't hesitate to chime in! Recommend what you like and let's encourage more exploration of the wonderful world of Gensokyo!