Obviously, the Starter Set. But when I see the catalog of my game shop I saw there's Dragonfire, Dungeon Masters Guide, and Monster Manual. But do I need all these to have an interesting D&D experience?
and welcome! This is a great question, and I'm going to assume since you're referring to the Dungeon Master Guide that you're talking about D&D, and since you're talking about current game-shop stock, that you're talking about 5th edition. If either of these things are incorrect, ignore this answer.
As a player, the only thing you truly need to start playing D&D is the Player's Handbook
As a DM, the only thing you truly need to start DMing D&D is the Player's Handbook
Wait, what? The same book? Yes.
Although the stuff in the DM Guide and Monster Manual will help with creating consistent adventures with appropriate loot and pacing (DM Guide) and save time having to create balanced monsters for your players to fight (Monster Manual), they are ultimately not necessary to play, as long as you are willing to create these things on your own.
As a side note, Dragonfire is a D&D-themed card game, so it has nothing to do with the tabletop game you might be familiar with, aside from sharing thematic elements. And the Starter Set only really gives you enough to play the basic campaign it comes with, it gives you a pamphlet-size version of the Player's Handbook (the Basic Rules) and a short mini-adventure with monsters and magic items as a substitute for the DM Guide and Monster Manual.
If you are truly new to the game and not sure if you'll like it, the Starter Set is a good buy. But if you already know you want to play, and want to hop straight into a table, get the Player's Handbook instead. The Starter Set won't have enough info to get you all the way into an existing group. Note however, that depending on your group (if they are people you know well already vs strangers in a game shop) there will be players there with copies of the Player's Handbook already. You might ask to borrow one to look over if you aren't ready to spend the money on it just yet. Many people get introduced to this game "on loan," using someone else's books. There's nothing wrong with that, whatever gets you to the table!
Best of luck!
You don't need to buy anything to get started with Dungeons & Dragons in general, though you may need to buy some stuff if you have a particular edition in mind and don't have any friends who own rulebooks you can borrow for that edition.
This answer will address what you need at a minimum as a player to make a character, and as a DM to run the game as recognizably that edition of the rules.
You need the Basic Rules, which can be downloaded for free from Wizard's website. There are some significant differences between the basic rules and the rules in the Player's Handbook and Dungeon Master's Guide (and between either and Xanathar's Guide to Everything), but the Basic Rules nevertheless create a game which is distinctly 5th edition.
Given that you are a new group, you might also want/need a premade module rather than developing a campaign on your own. You can get premade modules for 5th edition all over the internet for free. You would want a module designed for 1st level characters.
You need the Quick Start Rules, which can be downloaded for free from Wizard's website. You also probably want, as a new group and also because of the DMing instructions therein, the Keep on the Shadowfell adventure which can be downloaded for free from Wizard's website.
You need the System Reference Document, which can be accessed via any of a number of sites online, for example this one. Beware of dandwiki.com, which also has homebrew content new players often don't realize is homebrew. There's nothing wrong with homebrew content, but 1) the content at that website is often really bad, and 2) it's bad for learning a game if you conflate house rules and homebrew content with 1st party rules and official content.
Again, you will want a pre-made module. You can find them online, but a lot of the links are broken because Wizards published a bunch of adventures for free and then broke all the links when they redid their website. Finding fan-made modules is easier and less frustrating. Nevertheless, here is a link to an achived page wherein the adventure Wreck Ashore for 1st level characters was released (note the broken link), and here is a copy of that pdf that used to be linked there instead archived on some random other site I found via Google.
This Wikipedia article has a list of Wizards' free released adventures for 3.5 mixed with those for 3rd edition. It links to the archived announcement pages, but as you will have noted above, that doesn't actually get you a copy of the adventure. It does, however, let you know that that adventure's PDF was released to the public for free, and consequently you are on the legal side of copyright law in tracking down a digital copy of it.
Note also that, unlike the Basic and Quick Start rules above, the SRD is not designed to help somebody learn how to play but rather as a reference document for people making material for the system. It has everything you need, but it can be very obtuse to read.
Third edition had its own SRD before the 3.5 revision. You can also find that online, for example here. You can find pre-made adventures via the same Wikipedia page I linked earlier, or via Google. You can also generally use 3.5 modules for 3rd edition characters without issue, though the reverse is more tricky.
AD&D 1&2 e
You need books for AD&D; there isn't a system reference document officially released for free or anything like that, though that hasn't stopped people from putting the books or versions of the rules or retroclones of the rules online for free anyways. PDFs of the rulebooks are obviously copyright infringement and retroclones are obviously not. SRD-like "here's the rules without any original text" versions are a gray area where people disagree about what is and isn't copyrighted (e.g. does a creature's stat block constitute a particular artistic expression independent of the physical representation of that stat block, or is it more like a section of rules e.g. how many points an 'A' is worth in Scrabble?) and what is and isn't infringement.
In any case, you can buy either original books on EBay or some such, or you can buy the recent commemorative "premium reprint" versions, including possibly as a PDF for ~10$. You would want, at a minimum, the Player's Handbook and Dungeon Master's Guide for either single version of the rules. You would also probably need the Monster Manual, but maybe you can find a free adventure that includes its monster's stats and figure out how to read such a stat block on your own.
You can probably instead borrow a book to make a character from whatever group you join and thus do it for free. But without a group you'd need to buy the books.
I sometimes feel like you could play with just Skills and Powers, but that's probably an unfair characterization and you would certainly need to reverse-engineer a bunch of stuff.
You need the Basic rules, in one of their versions, which looks something like 5$ for a PDF from wizards. Again, you could make a character with help from a friend instead if you find a group that is using this edition, probably, which would make it free.
Original box sets D&D
They're basically collector's items at this point. You'd need a copy of or familiarity with Chainmail and a copy of Outdoor Survival if you want to use the wilderness rules, neither of which are D&D products. You could get by with just Men & Magic and Monsters & Treasure-- Volumes 1&2 of the set-- if you use the optional rules for Combat that became later editions of D&D.
I mentioned these earlier when talking about the internet presence of AD&D. These are technically separate non-D&D systems, but they often are basically indistinguishable from playing one of the early D&D games (i.e. AD&D 1e, 2e, or any version of Basic D&D. They are not very like the boxed sets). Often they are free, as is the case for OSRIC, but sometimes not, like Lamentations of the Flame Princess
Free Basic Rules are all you need to Get started
Honestly, you can get by with the Basic Rules only. The rest is gravy.
From what you mentioned in your question i assume you have no experience with the hobby so i recommend you use D&D 5th edition, as it's the latest and most simple version of the game.
If you just want to have a taste of the game, you can use the free basic rules available on the wotc website and character sheets also available for free. You can also find a free basic adventure on the Dungeon Masters Guild.
Now if you're serious about starting the hobby or just want to go all out from the get-go, you can either buy the starter set, wich includes basically everything you need to start. Or you can buy the Player's Handbook for in-depth general game rules, the Monster Manual for monsters stats and background, and the Dungeon Master's Guide for more advanced rules, tips in world-building and magical items, etc.
From there on there's a lot of other books of adventures, creatures, history, etc, for you to burn your hard earned cash on.