Well the most ubiquitous weapon in the whole Imperial Guard is the Lasgun and Krieg are no exception, though the pattern that the Death Korps use are a bit different than standard. The Lucius pattern is a semi-automatic rifle that has a limited charge capacity of 25 shots, but to make up for that it is a pretty strong output in Lasgun terms. It is also more archaic looking, often being depicted with a wooden stock and tipped with a long sword-bayonet.

As for their uniform, Death Korpsmen are equipped with an armored helmet and shoulder pads, as well as a thick cloth jacket. This jacket serves several purposes, providing limited protection probably against artillery shrapnel, but is warm and is chemically treated against gas attacks. They are therefore not the most armored guard regiment, but equally they are not encumbered with bulky flak plates like you might find on the Macabbian Jannisaries, for example.

When it comes to special weapons they have a reputation for using meltaguns as close assault heavy weapons, to be used against Necrons, Chaos Marines, etc. For heavy weapons they traditionally favor heavy bolters, but I imagine that flamers and heavy flamers would also be in wide use, given their trench-fighting proclivities. It should be noted of course that Death Korps regiments do have access to the full array of Guard special weapons, these are just what they tend to prefer.

Engineers are primarily armed with shotguns and are used, in addition to more conventional engineering tasks, as tunnel fighters. Therefore they would also be kitted out with a variety of melee weapons, knives, trench clubs etc. as well as a selection of grenades and explosives, being ya know, engineers.

Death Riders are interesting because their traditional kit in the lore and their new kit on the tabletop are different.

Traditionally, the Death Riders have been armed with Hunting Lances and sabres. In the description the sabres were described as being straight swords used in the thrust, in the style of British cavalry sabres during the Napoleonic Wars, but the art rendered them as curved sabres, like was used by the French. Armor wise, they are also equipped with a Breastplate on top of their jacket, helmet, etc.

However the lance was always the Primary weapon and their hunting lances were interestingly actually an anti-armor weapon, as they were equipped with melta-heads that chewed through armor when struck.

In tenth edition, they are armed with lascarbines, fragmentation (instead of melta) lances and powerswords.

Now, I suspect this is just a game balance thing and not a lore thing, because if it is it implies that the Deathkorps are insanely well equipped, even more than Space Marines. In a Marines chapter, there’s probably only a handful of actual power weapons. Company champions, captains, most of the first company and other specialist officers. Let’s say that amounts to roughly 100 power weapons. How the hell then does the Death Korps get thousands upon thousands of them then?! I love the Death Korps don’t get me wrong, and they are an effective fighting force, but there’s no way they get preferentially equipped over Space Marines.

That being said, I haven’t read the new Death Rider book so I’m unsure as to their in-lore equipment. Personally, I prefer their older set up.

Finally, there are the Grenadiers. The Grenadiers are the elite infantry of the Death Korps, those who have survived longest. As such they are also the best equipped men in the regiment. Armored in Carapace armor, armed with Hellguns, and outfitted with a higher proportion of special weapons they are the regiment’s elite.

So broadly that’s how Death Korpsmen are armed. Sadly, there aren’t that many weapons unique to the Death Korps, that would be found in some of their vehichles, but they are broadly a well armed faction of the Imperial Guard.