While there are plenty of gameplay-focused guides, there are no guides aimed at people who want to understand the story, lore and characters of Tsukihime and Melty Blood. This guide is aimed exactly at that: tell you what you should watch and/or read to get who are all these characters and what's going on in this game.

Before you continue reading, I should probably tell you this:
If you are the type of person who doesn't enjoy reading long visual novels, Melty Blood's story is probably not for you. However, if you haven't read any VNs yet, there's no harm in trying it out.

MBAACC is the latest installment in the (pre-remake) Tsukihime franchise. This means that if you try getting into it's the story with no prior knowledge it won't make any sense and leave you confused. While the game is fine as a stand alone fighter, you won't get much out of it story-wise if you haven't played through all the previous installments, which I'm going to list in their release order.

Keep in mind, that nearly all of them were never officially released in English, and are not even officially on sale any more. The original developer, Type-Moon, have long since moved on to their newer and more profitable projects. What I'm saying is that you aren't doing anyone any harm if you decide to borrow some of them. I won't give you any links, but you can easily find everything that I listed in the OP of Type-Moon General, over at 4chan's /jp/ board.

While it's not connected to Tsukihime directly, Melty Blood does feature KnK's protagonist, Ryogi Shiki as a fighter.
Both the original novels and anime movie adaptations are good ways to experience the story IF you speak Japanese. The original novels' English fan translation is of questionable quality. For an English speaker, the movies are probably your best bet. Please watch them in their release order.
If you end up not enjoying KnK, please still give Tsukihime a try. They are vastly different in how they present their story, so there's a high chance that you will still like Tsukihime even if you hated KnK.

Tsukihime is the first major release by Type-Moon, and the foundation of everything in the series. It features 5 heroine routes, with each having it's own unique story scenario. It's rough around the edges, but overall a fun read. It is important to clear every route before moving on to the next game. Arcueid's route has a manga adaptation, which is regarded as a good adaptation by fans.
Keep in mind that the Tsukihime Remake does not feature the same story as the original release, so if read one version, it doesn't mean that you can disregard the other. The remake also forms it's own canon and is not connected to the pre-Type Lumina Melty Blood games.
You can find a complete walkthrough here.[www.xerblade.com]

There is no Tsukihime anime.

PLUS+DISC is an extremely short (<2 hours long) collection of short stories connected to Tsukihime. It has some fun story bits and is not a huge time investment, so there's no point in missing out on it.

Kagetsu Tohya is a sequel to Tsukihime, which takes place 1 year after it's events and uses a unique structure to tell it's story. It is also notorious for being a ♥♥♥♥♥ to complete. Why can be witnessed with one glance at the flowchart.[i.imgur.com] It is roughly as long as one heroine route of Tsukihime.
If you care for 100% completion, just follow the walkthrough.[forums.fuwanovel.net] However, if you want to have fun uncovering this story on your own with no interest in seeing every single scenario, it's fine as well.

You have finally reached the games with Melty Blood in the title. Congratulations!
Melty Blood is a sequel/spin-off to Tsukihime, which takes form of a fighting game and features a lot of characters from it and Kagetsu Tohya.
ReAct has it's own story mode with branching story structure and arcade modes, while AC and AACC only feature their own arcade modes. While it may be confusing to understand what and in what order should be played, there's a complete guide to seeing the entire story of all of these games avaliable here.[forums.fuwanovel.net]

Melty Blood has some manga published related to it, with Melty Blood manga adaptation and Back Alley Alliance Nightmare being the main points of interest. It can provide for a fun read for the game fans.