The I Legio are the Praetorian Guard, the most fanatically loyal servants of the Caesar. How loyal, do you ask? During Julius O'Reilly's coup against his father, once they realized the odds of their master keeping his throne was hopeless, half the legio fought to the death - and the other half committed mass suicide rather than serve another master. They have only ever been led by the Caesar himself, or by the Imperator (the highest ranking member of the Marian military - it used to be the Marian head of state's title, but after the office of Caesar was created it became the Caesar's right hand). Their tactics, as befitting their primary role (protecting either the Caesar or whatever he needds held at all costs - traditionally Alphard or Illyria) are heavily oriented towards defensive warfare and entrenchment.
That may have been in the time of Sean or Julius, the I Legio being the most traditional, but from 3080 onwards they have been out of their once traditional location in Alphard.
Illyria is more the Base of operations of 2 Legions, the II Legio and the IV Legio, which alternately follow each other as a garrison of the Province + some merc units
It is more from the period of 3090 onwards they asked to be hunting pirates from the Ruins of Circinus from then on because the Legio felt responsible for the death of Cesar who was the father of Ignatius in his campaign in Logan Prime, after this they took as Base Valerius/Blantleff to make defensive actions, and offensive campaigns against the pirates of Circinus and beyond.
The two most Defense oriented Legions historically were the IV Legio specialized in Insurgency/Counter-Insurgency.
The III Legio before the last events was a Legio Limitanei, that is, a Legio for the defense of its own borders/territories, (Towards 3090 this changes).
The II Legio is the most offensive oriented legio, it is an Assault/Heavy Legio oriented to the planetary assault, if I remember correctly the II Legio was the Legio that during the period 3080/3130 destroyed a couple of Marik units (One of Marik's Protectors and I can't remember the other one).
Nominally Caesar does not have a Praetorian Guard (at least one unit of mechs that is), if there is an Infantry unit (Mobiliarius that is attached to the Custody of Caesar and the Senate)
This is something that always surprised me that a unit of "Bodyguards" of Cesar had not been formed, of at least a Reinforced Cohort, there were times when the Mortuori Cohort fulfilled that function (if quite strange, right?)