A couple of questions to start with:
I've only seen the first couple of episodes of the original show, but the implication (which sems to be confirmed by what people have already said here) is that basically even when untransformed Ichigo is an organic brain with a more or less wholey robotic body, with a layer of skin and hair over the top. So when he henshins, does the rider costume appear over his skin, or replace it? Either way I guess it's basically just a visual change.
What exactly happens with Black? It certainly seems like a body change, but is it a suit too? This may get answered in the show but I've only seen 15 episodes so far. The first episode has him transform into an insectoid creature, which briefly appears in the henshin sequence thereafter, which to me implies that Kotaro turns into that. But that's not the familiar Black costume. So does he further change and the costume is meant to be his actual skin? Or does he transform into an insectoid creature, which wears armor?
On to the Riders I've seen more of.
Amazon
A full body change. He seems to become a completely different species. He even has teeth in his mouth when transformed in the earlier episodes.
Shin/ZO/J
Full body changes.
Kuuga
Full body
Agito, Gills, Another Agito
Full body transformations.
G3, V1, G3-X, G4
Suits
All Ryuki Riders
To me they are like biohazard suits. Humans seem to dissolve if they enter the mirror world. In blank form, the suits seem to delay the process for a limited amount of time. When connected to a mirror monster by a contract card, the suit takes on whatever property of the monster allows it to survive. I don't think it changes the person inside at all (otherwise Shinji would actually be growing a dragon head on his hand every time he used his strike vent card, and I never got that impression).
All Faiz Riders
I never got the impression they were anything other than suits.
Blade, Garren, Leangle, Glaive, Larc, Lance
They get called Rider Systems. They're suits powered by draining the energy from a sealed Catagory Ace Undead.
Chalice
Chalice doesn't henshin into a Rider. He has some kind of built-in ability to scan the card of a sealed Undead and match it's appearance. He's a shapeshifter. It just happens that the Mantis Undead looked quite "ridery".
All Hibiki Oni
A slightly confusing one. It's set up that they actually become demons. Hibiki demonstrates an opening mouth while transformed in some episodes. On the other hand, that makes it seem a bit weird when they transform just their head back to normal (how do the blood vessels and nerves line up if they're different species?), plus the tech girl introduces herself as the person who invented their henshin devices (even though the series otherwise insists Oni have been around for hundreds, if not thousands, of years). Personally I go with body change. It just seems to fit better with the rest of the series.
All Kabuto Riders
Suits all round.
All Den-O Riders
Suits. In the series it always seemed like Ryotaro gained abilities from being possessed by the Imagin. The henshin just covered him in an armor suit. When transformed but unpossessed he has no extra abilities over his usual self. None of the other characters seem to exhibit extra abilities from transforming either; they just gain armor and weaponry.
The one interesting bit to note is that what happens to the Imagin involved seems to vary. In their personal forms (Sword, Rod etc) they seem to just possess Ryotaro's body inside the suit. But in Climax Form some of the dialog suggests that they become the armor, as they react in pain when the visor representing them is hit. Similarly, New Den-O is meant to use the same henshin belt, but with him it turns Teddy's body into a sword. Deneb also seems to bodily transform when involved in a Rider form, becoming part of the armor or a weapon. None of this seems to affect the human wearing it though.
Kiva, Dark Kiva, Saga
Maybe both? Kiva's right leg seems to suggest that it transforms his body into something else (it can't just be Wataru's full Fangire self, as it's the same process that the Fangire King uses to become Dark Kiva and he's already his full Fangire self) and then covers it with armor. Some of the associated fluff seemed to suggest Kiva's normal form is wearing "the chains of destiny" that protect Wataru's human soul from being destroyed by his monstrous body in his transformed state. Garulu form's finisher shows that he has an opening mouth. I think the same also applies to Dark Kiva (like I said, it's the same process) and Saga (his belt is meant to be another monster species).
Ixa and Rey
Suits. Both transformations come from technological creations created by human organisations. Also, I think a device designed to turn human DNA into something else would have gone horribly wrong when trying to turn Jiro into Ixa, and it didn't.
Arc
Erm...not enough info. It's a mechanical belt rather than a monster-triggered transformation, but I can't recall if we saw Sugimura's Legendorga form, so he could be reverting to that and just having a Rider-like suit created over the top.
Decade and DiEnd
I vaguely recall some of the fluff treated them as suits and pointed out technological details, such as those black bracelets and anklets that are supposedly what gives DiEnd his fast speed. I never got the impression they were more than just suits.
All W Riders
I'm in the full body change camp. The stated function of all Drivers in the show is to filter the output of Gaia Memories so they don't warp the mind of the person using them. Otherwise the memory just does the same thing to them as it does to a Dopant. If you look at the Dopants, the Zone Dopant clearly wasn't a suit, as it was non-humanoid and too small to fit the user inside. When the Beast Dopant was hit by CJX's sword, deep gouges were created in the front of the costume, and he clearly states that he's been injured, rather than having damaged armor. If becoming a Dopant is a full body change, then so is becoming a Rider, as they are essentially the same process.
OOO and Poseidon
More complicated due to the alchemical origin of the medal powers. The OOO costume design would suggest a suit rather than a full body change. However, the fact that the cheetah medal allows him to move faster, and ShaUTa can become a liquid, would imply something beyond just armor. Some medals give powers that imply a simple suit change, some a body change, and some could be either. I suspect the designers didn't really think about it tbh.
Birth/Proto-Birth
A suit.
Aqua
Not enough info in the film
Fourze and Meteor
Suits
Nadeshiko
She seems to henshin like Fourze, but given that she has shape-shifting abilities anyway I suspect this could actually be a full body transformation.
Wizard
Only five episodes to go on, but after the "error"/out-of-mana sequence at the end of episode four and start of episode five I'm tempted to say suit.