How many seasons are there in the Fate anime series ?

Each fate show has 2 seasons. Although you will most probably find it by its episode numbering rather than seasons. Seasons dont make much of a difference watch it as a single show.

Recommending the order of watching the series is a very tough task. The Visual Novel on which the show is based works on the principle of parallel reality. In simple words the premise of the story is the same yet it unfolds in different ways. There are 3 arcs... 1.) Fate stay night 2.) Unlimited bladeworks 3.) Heavens feel

Though the start of all 3 are similar the story unfolds completely differently. Moreover there is a prequel show called Fate/Zero.

The problem here is that Fate/Zero reveals a lot of plot twists and can prove to be a spoiler for the rest of the 3 arcs. At the same time watching Fate/Zero first adds a special dimension since its chronologically correct and you know what happened 10 years before which has brought these character together. That knowledge is special.

Its your call entirely how you want to watch the show.

In the Fate Series, there are multiple realities or parallel universes if you will, each with its own stories to share. There are many different stories in the Fate universe, mainly:

The Main series

This is the original series that first had an anime adaptation in 2006. It was also the origin of the now legendary meme, “People die if they are killed”. It consisted of three different routes: the Fate Route(Saber/Artoria), Unlimited Blade Works route(Rin), and Heaven’s Feel route(Sakura). Each route is a what-if situation where the main heroine is a result of Shirou’s choices during the beginning of the story. There is also a prequel, Fate/Zero which explains much of the past of the various characters and gives extra insight. The order of watching this should be:(Watch the original 2006 series if you like Saber and would like to see her as the main heroine. The animation looks dated now and does not compare to the excellent animation quality of the newer ones)(You may watch Zero later too since it does give away some key spoilers)

1. Zero(2 seasons, 12–13 eps each)
2. Unlimited Blade Works(2 seasons, 12–13 eps each)
3. Heaven’s Feel(3 movies)
4. Carnival Phantasm(Parody specials, highly recommended, 12+ episodes and counting)
5. Hollow Ataraxia(Epilogue)(In the works. Will probably come out after all 3 Heaven’s Feel movies have been released)

The different Grail War stories

There are many alternate universe Fate storylines each with different heroic and not-so-heroic Servants and different masters and protagonists in different settings. Each of them has nothing much to to with the main series and can be watched separately without any issues. Currently the following have/will have anime adaptations:

1. Grand Order(Specials, 1 released, 1 more coming soon)
2. Apocrypha(1 season, 24–26 episodes)
3. Extra(1–2 seasons, coming soon)

There are also several other stories happening in the same universe(Nasuverse), but they are not about Grail Wars. Thus, they are not Fate, but happen in the same universe, so I’ll just mention them in case you wish to watch them:

Kara no Kyoukai

A movie series about the avatar of Akasha, Shiki Ryougi as she controls her desire to destroy everything that exists and settle into normal life. As an investigator of seemingly paranormal activities, she tries best to balance life and work, with varied success. Each movie is a work of art is are highly recommended. The plot is however, a little difficult to understand, but if you have watched Fate, the terminology will become instantly familiar. There are 7 movies to be watched in airing order, namely:

1. Fuukan Fuukei
2. Satsujin Kousatsu
3. Tsuukaku Zanryuu
4. Garan no Dou
5. Mujun Rasen
6. Boukyaku Rokuon
7. Satsujin Kousatsu Part 2

Tsukihime

A story about the boy Shiki Tohno and his meeting with the true Vampire Princess, Arcueid, from which all other vampires and True Ancestors have originated. Shiki by reflex, cuts the Vampire Princess into a million pieces, weakening her heavility and making her fall in love with him(logic?). Then, they team up to investigate various murders and mysterious happenings in the city. The victims have their blood sucked dry. Can this all be Arcueid’s doing? He also has to deal with various other characters like his sister Akiha, the maids Hisui and Kohaku, his upperclassman Ciel, classmate and good friend Satsuki and fights various extremely powerful foes.