1. Goals
Hadrianus has the following general goals, which will need to be taken into account when deciding which activities he will undertake.
First, develop his new covenant. This is something he can get to work immediatly. Having a strong covenant is valuable in itself and the right thing to do as a service to his sodales and to the Order. But it will also be useful as a pilotical tool to further his other goals. Exactly how to develop the covenant will depend on its location and setting. Although Hadiranuas may very well become the covenant's leader because of his natural aptitude in Leadership, he will contribute regardless of his actual position in the covenant's hierarchy. This includes building and training with the covenant's turb, establishing and maintaining the covenant's resources, or whatever else he is able to do.
Second, rehabilitate his pater's name. This will be difficult to do from a distance, but correspondance may be a good way to start. Hadrianus would like to discover who was behind the plot, but lack of resources and contacts made that impossible so far. Thus, establishing a base of operation and (re)building his credibility and contacts will be a first step towards that. He also plans on writing a series of books collected under the name of "The teachings of Traianus" covering various subjects that his pater introduced him to (such as leadership and the art of Perdo).
Finally, make a name for himself. Again, how this will materialize will depend on the context of the saga (setting, Tribunal, mundane and magical landscape). This could lead him to develop his combat capabilities (for a role as a Hoplite), his political skills (to achieve a significant role in the Tribunal's hierarchy), find a noteworthy familiar or discover some lost treasure. He will start small, since he knows he is currently in a position of deficit, credibility-wise. But he will always keep an eye on higher steps and is confident of his abilities.
2. Abilities
First, Hadrianus needs to establish himself at his new covenant. One of the things he must quickly learn is a local language, as well as some Area Lore. This will help him interact with locals and search for resources.
Second, improbing his combat effectiveness is someting important, so he will seek to improve his scores in Parma Magica, Penetration, Concentration and various spell masteries. Learning how to handle weapons would also positive, probably for the use of a staff, thus the Great Weapons.
Third, interaction with the covenant's turb will require improvement in Leadership. That will also be useful in establishing better relations with other magi of the Tribunal. Other social skills will be useful there as well, so Bargain, Etiquette, Intrigue and Guile are also on his list of priorities.
Finally, furthering the legacy of his pater will require him to be able to improve himself magically, so improving his knowledge of Magic Theory, Code of Hermes, Organization Lore: Order of Hermes and the various Realm lores (mostly Magic Lore) will be useful.
3. Spells
There is a great many spells that Hadrianus will want to learn or invent from scratch. These can be categorized under combat spells, utility spells and information-gathering spells. Combat spells can further be qualified as offensive and defensive spells.
His current offensive spells (Parching Wind, Grip of the Choking Hand and Blade of the Virulent Flame) are already pretty good, particularly combine with Flexible Formulaic Magic and his spell masteries. But there might be additional offensive spells that might be useful for some special situation. Opening the Intangible Tunnel (ReVi gen) is a spell with good offensive potential, which should be mastered both for penetration and magic resistance. Dust to Dust (PeCo 15) might be a good idea, particularly with a necromancer amongst the covenant members, and is a spell that he can invent in a single season if the aura is 3 or more. An instant-kill spell such as Clenching Grasp of the Crushed Heart (PeCo 40) is currently out of his reach but eventually he will want to learn one. Other specialized spells for the various forms might also be useful, but these will depend on the covenant's surroundings. There are many spells that might be useful there, such as Creeping Oil (CrAq 15), The Wound That Weeps (PeCo 15), Invocation of Weariness (PeCo 20), Incantation of the Milky Eyes (PeCo 30), Twist of the Tongue (PeCo 30), Plant's Withering Bane (PeHe 20), The Great Rot (PeHe 25), Heat of the Searing Forge (CrIg 10), Winter's Icy Touch (PeIg 10), Pit of the Gaping Earth (PeTe 15) and a variant of Demon's Eternal Oblivion (PeVi gen). Many of these spells might be mastered for Magic Resistance, rather than actually used for offensive purposes.
Two of his current defensive spells (Curse of the Rotted Wood and Rusted Decay of Ten-Score Years) protect him from common mundane threats. His third spell (Veil of Invisibility) is more passive. All of them offer good flexibility for other uses. Other spells that he could learn to cover other Forms and other situations include Decay Fur and Hide (PeAn 10), Quiet the Raging Storm (PeAu 20), Endurance of the Berserkers (ReCo 15), Wizard's Leap (ReCo 15), Soothe the Raging Flames (PeIg 20), Fist of Shattering (PeTe 10) and Stone to Falling Dust (PeTe 20). Against magical threat, various versions of Unravelling the Fabric of {Form} (PeVi gen) and a more general Wind of Mundane Silence (PeVi gen) would both be useful. Of course, all of these spells should ideally be mastered for fast casting.
Information-gathering spells that might be useful include True Sight of the Air (InAu 15), Discern the Images of Truth and Falsehood (InIm gen), Dispel the Phantom Image (PeIm gen), The Invisible Eye Revealed (InVi gen), Sense of Magical Power (InVi 2), Scales of the Magical Weight (InVi 5) and Sense the Nature of Vis (InVi 5). Most of these are low-level spells, since his knowledge of Intellego is ono-existent at the moment.
There is a wide array of utility spells that might appeal to him, depending on the location of the covenant. These would include Comfort of the Drenched Traveller (PeAq 5), Cloak of the Duck's Feathers (ReAq 5), Chamber of Spring Breezes (CrAu 5), Purification of the Festering Wounds (CrCo 20), Eyes of the Cat (MuCo 5, An), Rise of the Feathery Body (ReCo 10), Lifting the Dangling Puppet (ReCo 15), Lamp Without Flame (CrIg 10), Dispel the Phantom Image (PeIm gen), Words of Unbroken Silence (CrMe 10), Trust of Childlike Faith (PeMe 10), Snap of Awakening (ReMe 10) and Unseen Arm (ReTe 5).
4. Talisman
A talisman is a good way to help Hadrianus boost his magical capabilities, particularly with regards to penetration, but also for effects to be used in deleterious auras. A wand or staff would make a lot of sense because of its basic shape bonuses (+4 to control/destroy things at a distance, +3 project bolts and other missiles, +2 repel things). Materials of interest are Hornbeam, Topaz, Ruby and Silver. Further thought will be required before he undertakes that project.
5. Familiar
Although Hadrianus does not have immediate plans to bind a familiar, depending if the opportunity arises he will do so. So, if a suitable creature is discovered near the covenant or during an outing, he will investigate it to determine if it is suitable for him.
He has no specific idea at the moment, but RoP:M p.57 suggests the following animals as having an affinity with the art of Perdo: "asp, hedgehog, salamander, shrew, weasel, any venomous animal." I don't see his familiar being the companion-level character of another player, but am not adverse to the idea
Using his current stats, his Familiar Bonding Lab Total would be 26 + aura (Pe 17 + Form (An/Aq/Co/Ig) 4 + Int 2 + Magic Theory 3). Since his form scores are so low, this lab total could increase quickly with a few seasons of study.