This whole relationship of Central Asian groups throughout history and the lack of sources just perplexes me, and the relationship they share is dauntingly complex. To my knowledge, the Bulgars are a cousin group to the Turks, yet speak a Slavic language. There is also the oft mentioned Turco Mongol tradition of Central Asia, and obvious cultural ties, but they are classified as distinct groups. Were they once one group in the past or were they always separate and simply diffused culture significantly? And how do the Huns, Hungarians, and Xiongnu fit into all this, as I've heard they were one people, three completely separate people, or distant cousins?
EDIT: This is just an add on I thought of later, as I recently recalled as well the Tartar people, along with the distinct Tatar people, and the geographic regions of modern day Tatarstan in Russia, Great Tartary, and Chinese Tartary. To my knowledge, the Tatars and Tartars are distinct classifications, and the Tatars of Russia and Ukraine are Turkic. How they relate to the Tartars, I don't know, and I have also read that the Tartars were a Mongolic people, but also Turkic in different sources. Then there's the Manchus, the native people of Manchuria in North Eastern China, whom I see often thrown into all this as part of Chinese Tartary.
Usually when we talk about relationship between different people we mean two thing: linguistic and genetic relationship. So first take a look at these, and then I will tell you about Hungarian ancient history - as I have very limited knowledge about the ancient history of the other people you mentioned.
So linguistic relationship. This is a very-very distant relationship for the people you mentioned as a whole. As you yourself say, Bulgars today speak a Slavic language, which is Indo-European, Hungarian belongs into the Finno-Ugric group, while Turkic and Mongolic languages belong to the Altaic group.
If you are wondering about genetics, then the difference is at least as big. Hungarian population for example is closest related with Slovaks and Austrians if you are looking at Y-chromosome haplogroups.
With that out of the way, let's speak about us, Hungarians.
The presumed Finno-Ugric ancient home lies in the Uralic mountains, and around it. Here Finno-Ugric groups were practicing a fishing-hunting-gathering lifestyle. It is estimated that the breach between Finnic and the Ugric branch has happened between the XXVI. and XXIV. century BC. It was a time of climate change, and the Finnic people followed the line of the deciduous forests as it retreated towards the West, while Ugric people stayed where they were/migrated South/East, and changed their lifestyle to one that fitted better to the groved steppe.
That is the time when Ugric people would have started to rear domesticated animals, among other horses. Vocabulary in Hungarian regarding husbandry relates to that of Turkic and Iranian languages, which means that animal husbandry was probably learnt from these people: Iranian people more or less residing were they are now, and Turkic loanwords must have come form the Turkic people then living in today Kazakhstan.
The ancient Hungarians broke of from other Ugric people in the XII. century BC. The Obi-Ugric branch has two living languages today, the Mansi and the Khanti, and sadly both are likely to die out in one or two generations, as Mansi has under 1000 native speakers, while Khanti has under 10 000.
The older scholarship assumes that Hungarians lived somewhere east from the Ural river in this era, in today Kazakhstan, and slowly, generation by generation migrated westwards. New archeologic (Sudár Balázs) findings suggest that we spent much-much longer time in the East, and then moved to the Black Sea in a short time possibly in less than a century. Also, in the former case we were assumed to be a vassal people of the Kazar empire, while in the latter case we were cohabiting with other peoples and polities.
Whatever the truth is, the first surviving written source which mentions Hungarians is generally thought to be Constantine VII's De Administrando Imperi at the beginning of the X. century. To add to the confusion, he calls us "The Turkic people", because why of course he does.
Hungarians became involved in the European affairs the usual way: a former steppe people, the Bulgars were the current nemesis of the Empire, so the Romans (or Byzantines, as you like) hired the next available migratory people to fight them. From what I gathered from Ostrogorsky this horribly backfired in 80% of the cases, and only resulted that the new migrants became an even more formidable neighbour of the Empire than the former ones, possibly even swallowing imperial territories here and there. But we were one of the lucky exceptions, because we really did annoy the Bulgarians without wiping them out, and set up shop in the Carpathian basin. We did send expeditions in imperial territory later, and did pillage/burn most of the Balkans, but we equally annoyed the Bulgarian empire, and also spent a lot of effort into pillaging German lands in the next decades. When in the XI. century Basileios II. came around to destroy the Bulgarian Empire, we even proved to be a more or less useful ally against this old enemy.
And now to alternative hypotheses. László Gyula, a prominent archeologist has two very unique theories about Hungarian history.
First, he says that the common vocabulary of Hungarian and other Finno-Ugric is too little to justify the common ancestor theory, and he further speculates that this common vocabulary comes from a sort of common trading language of few thousand words that these people developed while they cohabited in the Uralian region. In this case ancient Hungarian would be a language family on its own, unrelated to any other living language today.
The second theory says that old Hungarians were actually the Pannonian Avars, and arrived in the VII. century into the Carpathian basin. In this case the people who arrived into the Carpathian basin in the X. century really were a Turkic people, and the two people fused much like old Bulgars and Slavs, creating the current Bulgarian people, or like Anglo-Saxons and Normans melting together into the English people. This theory would also neatly explain how there are also Turkic grammatical elements in Hungarian, and not just loanwords. Or you could say that those grammatic elements are result of previous Turkic influence. Whatever let's you sleep at night.
After reading this far you might wonder where the famed Hun-Hungarian relationship comes into the picture. The answer is: we have no idea at all. It is true, that in the oldest Hungarian gesta, the Gesta Hungarorum the unnamed author says that Hungarians and Huns descend from a common ancestor, but we have no idea if the author founded this statement on oral, or earlier written sources, or just made it up. Furthermore, we can only speculate what kind of language the Huns spoke, so linguistics cannot help to establish a relationship either. We are either related to the Huns or we are not, but it is advisable to tread around this question very, very carefully, because usually in Hungarian academics you make yourself into a laughingstock, if you bring up the Hungarian-Hun relationship as a serious theory.
One last note: there is an interesting question in Hungarian history; the existence of Magna Hungaria. Supposedly, when the Hungarians moved into the Carpathian basin, not the whole people moved, several of them remained back in "Greater Hungary" wherever it has been. The matter of fact is that we have three reports about a Hungarian speaking people in Volga Bulgaria.
The first, Julianus, a Hungarian friar travelled in 1235-36 to the East, and found these remaining Hungarians. He returned to Pannonian Hungary thereafter, and next year he travelled back to Volga Bulgaria to set up future missionary work, and to start diplomatic relations with these Magna Hungarians.
The latter case is exceptionally interesting, because the ruling king, Béla IV. had this pet project, that he wanted to settle nomadic people in Pannonian Hungary to have his very own horse-archer Praetorian Guard against both outside enemies and unruly nobles. He later realised this project with Cumans, and it worked out surprisingly well, but the start was rocky, and for the first time it actually resulted in a Cuman-Hungarian civil war on the eve of the 1241 Mongolian invasion. One can wonder whether or not it would have worked out better if Hungarians of Magna Hungaria would have been at hand.
Anyways, when friar Julian reached Volga Bulgaria in 1237, the Hungarians of Magna Hungaria have been already wiped out by the Mongolian Empire, so the second report of Julian only relates these sad news.
The third report of Magna Hungaria comes from the pen of a papal legate, Plano Carpini, who travelled to Karakorum in 1246. He briefly mentions the Mongolian conquest of Magna Hungaria, and the destruction of its people.
Sources:
László Gyula: A kettõs honfoglalás, Budapest, 1978
László Gyula: Árpád népe, Budapest, 1988
Csepregi Márta (ed.): Finnugor kalauz, Budapest, 1998, Panoráma
Petkes Zsolt, Sudár Balázs (ed.): Magyar õstörténet 4., Budapest, 2016, Helikon
EDIT: I realised, that while I answered the letter of your question ("what was the relationship") to the best of my abilities, I didn't tackle the soul of your question ("what is the connection"). So the connection between these people you listed is that all lived on the Asian steppes at a certain point in their history, and also all of them were nomadic/half-nomadic horse riding people around the same time. Some are truly related to each other linguistically/genetically on top of that, but not all of them.