There are a few "next steps" that you can attempt -

4th Edition D&D is, in many ways very, very similar to the D&D Adventure System games. I recall seeing some threads on the WotC forums to this effect, and http://www.dndboardgame.com/ appears to be an effort to convert some of the earlier Adventure System games to that version of D&D

You could try one of the similar boardgames such as Descent, which has a richer character advancement mechanic than D&D Adventure System, and has a "Dungeon Master" controlling the monsters. However that would be a potentially expensive investment, and doesn't really bring you much closer to an RPG experience.

You could try an RPG like the Warhammer Fantasy Roleplay - https://www.fantasyflightgames.com/en/products/warhammer-fantasy-roleplay/ - which puts less emphasis on a pencil and paper character sheet, and rather has a feel closer to the Adventure System games with cards and other tactile elements.

A key difference is feats, but class options are also limited.

While there is considerably less content in the SRD than the PHB for sub classes, spells, backgrounds, and feats, there being only 1 of 42 feats is a significant difference.

Feats

Feats are an optional / variant rule in the "customization" chapter of the Players Handbook (Chapter 6). There are 42 of them, any of which can be chosen in place of an ASI at level up at 4th, 8th (and so on) level. The SRD provides one: Grappler. (Based on the discussions I've seen on CharOp boards, it isn't such a great feat and is of little use to most classes).

Related to feats is the lack of Variant Human option

The variant human is a popular choice since rather than getting +1 to each ability score, it allows the player to choose +1 to two ability scores, one feat, and an additional skill.

What this attempts to do is address the matter (among other things) of the variety of race based benefits (like darkvision) that makes the other player races more capable/powerful at lower levels.

Backgrounds

The rest of the backgrounds -- a total of a dozen (PHB p. 127-141) -- are missing as well as the skills/abilities/tools that accompany them. The Basic Rules provide 4 backgrounds (Sage, Soldier, Criminal, Acolyte) and the SRD only one (Acolyte).

Spells

The SRD and Basic rules do not contain the complete list of spells in the Players Handbook. Of note, missing is the Warlock spell Hex, and numerous "Paladin Only" spells. (Only Branding Smite is included, while there are multiple smite spells (charge ups) in the PHB.

Sub Classes/Archetypes

The SRD and Basic Rules only offers one sub class each for the player classes. That's another significant reduction in content. Most classes have three build paths (though sorcerers only have two, Clerics have sev, and Wizards eight Arcane Traditions: one for each school of magic).

For example: the SRD only presents the Oath of Devotion for Paladin (missing Oath of Ancients and Oath of Vengeance) and for Clerics only offers Life Domain (missing Light, Tempest, Knowledge, Nature, War, and Trickery). It also lacks the Eldritch Knight (Fighter) and Arcane Trickster(Rogue) spell casting options for Fighter and Rogue classes.

Monster Manual Detail, and missing Iconic Monsters

There is a far deeper description for each monster. Likewise, a lot of lore and variation is missing, as are some monsters.

• For example, the pseudodragon in the MM (p. 254) includes the variant "pseudodragon familiar" option which is not in the SRD. This is different from a familiar found with the spell.

• Some monsters in the Monster Manual are not in the SRD. For example, Mind flayers are not in the SRD. Beholders are not in the SRD.

  On page one of the SRD, the "product identity" monsters (whom you won't find in the open material) are: beholder, gauth, carrion crawler, tanar’ri, baatezu, displacer beast, githyanki, gith erai, mind flayer, illithid, umber hulk, yuan‑ti. (p. 1, SRD 5.1) This can be confusing (and require some DM adaptation) when you find reference to the beholder in the Deck of Many Thing in the SRD, but that monster itself is absent. (Note: there is some yuant-ti material (p. 22) but no lore/background, in the Hoard of the Dragon Queen supplement available for download at the WoTC web site).

DMG Customization options.

Aasimar (an angelic version of the Tiefling, DMG p. 286-287) and the Oath Breaker Paladin (p. 97 DMG), for example, are optional player character classes in the DMG that you don't find in the SRD. The SRD has a tan text box explaining the penalties that may arise from breaking an oath for a paladin, but it does not offer the Oath Breaker Paladin archetype as the DMG does. One feature of the DMG that is well laid out is the thought process that needs to go into creating custom races.

Having bought the DMG for AD&D 1e, 2e, 3.5, I can say that the 5e DMG is different from some earlier versions. It's light on rules (the game engine is mostly in the PHB) and heavy on advice for DM's. It has a lot of very good advice on how to run a campaign, ideas on settings, and a variety of optional rules (like facing in combat) that have been used in other editions.

• The DMG includes epic boons (p. 232 DMG) as a way to increase character capabilities when all of the ASI's and class features have run out after reaching level 20.

Is there enough stuff in the SRD and Basic Rules to run a campaign? Yes.

WoTC implied that there is enough material in the Basic Rules to run a campaign as far as 17th level (or higher) by including monsters of difficulty up to CR 17 (Adult Red Dragon) and level information beyond that.

There is some other free material available from WoTC via download.

Our group particularly liked the supplemental/free official material from the Elemental Evil campaign, and the Horde of the Dragon Queen/Tyrrany of Dragons campaign.

It's not the books, it's how you use them

We ran years-long campaigns with OD&D (1970's, before AD&D 1e came out) with a lot less material than is now available for those who play D&D 5e. (Quite frankly, the current material is far better organized). All that is needed is:

1. Players
2. A DM interested in keeping the campaign going and providing new challenges as the players progress in levels
3. Funny shaped dice
4. A bit of imagination.

Why I recommend buying the books

(Full disclosure: I am not a WoTC nor Hasbro employee, I own no stock in Hasbro)

1. All of the content and thus all of the player character class choices and options.
2. Better art
3. Better lore
4. Per @CM_Dayton's comment: "Supporting the authors / editors / artists."
   Item 4 isn't a short term benefit to your game, but it helps keep the game alive in the publishing world. (At least in small part).