What is the code on putting a ground rod at a transformer?

It depends on whose transformer it is. A utility owned pole or pad transformer is grounded under the utility rules (NESC), not the NEC, and the utility handles its own electrode. For a customer owned transformer, typically a dry type unit creating a new 208Y/120 or 480V system, the NEC treats the secondary as a separately derived system and 250.30 governs.

The electrode is not automatically a rod

NEC 2023 250.30(A)(4) requires the grounding electrode for a separately derived system to be the nearest available option in this order of preference: metal water pipe within the area served, or structural metal building steel that qualifies as an electrode. A driven rod is what you use when neither of those exists nearby. Plenty of installers drive a rod at the transformer out of habit when the code actually wanted a connection to building steel ten feet away.

System bonding jumper and conductor sizes

The system bonding jumper connecting XO to the enclosure is sized from 250.102(C)(1) based on the derived phase conductors, and it is installed at either the transformer or the first disconnect, not both. The grounding electrode conductor is sized from Table 250.66 using the same derived conductor size. If the electrode is a rod, the GEC never needs to be larger than #6 copper per 250.66(A). Do not forget the supply side bonding jumper with the primary feeder per 250.102(C).

When a rod is the right answer

Outdoor pad mounted customer transformers, remote structures, and buildings with no accessible water pipe or qualifying steel all end up on rods. Then 250.53(A)(2) applies: one rod must meet 25 ohms or be supplemented with a second rod at least 6 ft away. Bond it, protect the conductor from damage, and use listed clamps rated for direct burial where buried.