On the battlefield.
Something I love doing in Fire Emblem: Awakening is leveling up and reclassing my units, especially with the ones I like to use a lot. It should come as no surprise that Chrom is one of them. Because Chrom is one of the main characters he'll be level 1 to start. At this point in the game you will not be able to freely roam the map yet and therefore cannot summon Risen for grinding purposes. Because of this you'll have to be mindful of what you do with him so that you don't lose him, as, unlike most other units in the game, your game will automatically end if he or Robin dies.
Chrom's base class is Lord, which is unique to him and Lucina. As such there are certain skills only he and Lucina can learn. As a Lord Chrom can learn the skills Dual Stirike+ and Charm. Dual Strike+ increases the dual strike chance by 10% and Charm gives a 10% bonus to accuracy and avoid for all units within three squares of Chrom. I can see Charm being useful in the beginning, but not so much later on as my units tend to be spread out a lot more and perhaps even a bit overleveled. As such I tend to swap it out for something else as soon as I can. Once Chrom reaches level 10 he can promote into a Great Lord. It is a good idea to spend some time in this class early on as both the skills he learns are super useful. Aether combines the effects of both Sol and Luna (in that order), and Rightful King increases skill activation by 10%, making skills like Aether and Aegis activate more frequently.
Chrom's reclassing options are Cavalier and Archer. After I am done with Great Lord I will reclass Chrom into Cavalier for the Discipline and Outdoor Fighter skills. Discipline is great for leveling Chrom's weapon skills faster and Outdoor Fighter will increase his hit and avoid by 10 when fighting outdoors. Once I have those skills I will promote Chrom into either a Paladin or Great Knight. Both classes have useful skills so they are all worth picking up. I like to get the Aegis skill from Paladin as it halves damage received from bows, tomes, and dragonstones when it activates, which is especially useful as Chrom's resistance growth can be pretty slow. With the Rightful King skill this skill will activate more often making it even more useful. Luna and Dual Guard+ from Great Knight are also worth getting.
Aside from the fact that Chrom cannot use Falchion as an Archer or Sniper, there aren't really any super useful skills from these classes which would benefit him as a Great Lord––the only exception being maybe Hit Rate +20 from Sniper, but the only time it is really useful is when you're playing on harder difficulties. If you aren't then you're better off with something else, as, if you're anything like me, you will likely outlevel most maps making the extra hit kinda wasted. I will sometimes reclass Chrom to Bow Knight for the Bowbreaker skill, which gives him +50 to hit and avoid if the enemy has a bow equipped, but even then Aegis is a far more superior skill to have. Unless you plan to keep Chrom in one of these classes I would personally skip them altogether.
When I have all the skills I want I'll reclass Chrom back to Great Lord. Because you're only allowed five active skills at a time it can take a bit of figuring out. Skills I like to have on Chrom include Dual Strike+, Aether, Rightful King, Dual Guard+, and Aegis. Because I like to take advantage of the pair up system this set works out really good. If you don't plan to take advantage of it you'd likely want to swap out Dual Strike+ and Dual Guard+ for something else as they have no effect outside of pair ups.