On the battlefield.
One of the fun things about Fire Emblem: Awakening is leveling up and customizing your units in order to make them powerful fighting forces. Unlike the parent units the child units can require a bit more planning and probably a longer explanation on this page. Because of this I am only going to focus on one possible mother for Lucina: Sumia. Reason being is I have all the supports for Chrom and Sumia and so I have the most experience with Sumia as Lucina's mother as a result. I will also admit that I support Chrom and Sumia as a pairing. Anyway...
Lucina's starting class is Lord and she will be level 10 when she joins Chrom. While with most other child units you can level up and reclass their parents to get the desired skills to pass down to their children, Lucina is not so easy to do this with due to her mandatory recruitment at the end of Chapter 13: Of Sacred Blood. Unless you can somehow level Sumia to level 15 as a Dark Flier to get Galeforce before then (and if you can kudos to you) you may as well just stick to one of the other two skills Sumia has as a Pegasus Knight, which are Speed +2 (increases speed by 2) and Relief (Recover 20% of HP if no units are within a 3 tile radius). She can relcass into all the classes of both her parents anyway so it's no big deal what skill you give her. Lucina will always inherit Aether from Chrom even if he does not yet know the skill or has a different skill set as his last skill.
As a child unit Lucina has more reclassing options than parent units as she will inherit classes from both her parents. From Chrom she will inherit Lord, Archer, and Cavalier and with Sumia as her mother she inherits Pegasus Knight, Knight, and Cleric. After I get Lucina to level 20 I like to promote her to Great Lord first. Although Lucina will already have Aether from this class I like to get the Rightful King skill as it increases skill activation by 10% which is amazing when combined with Aether. Once I have that skill I like to reclass Lucina to Dark Flier for Galeforce which allows her to move again if she kills a target. After I have that I'll reclass her to either General or Paladin. I generally don't like General because the class is slow but the Pavise skill, which reduces damage taken from damage inflicted by swords, lances, axes, monster weapons, and Beaststones by half when activated is nice to have. Aegis from Paladin is also a good choice to have as it halves damage from bows, tomes, and Dragonstones when activated. Both are good to have in combination with Rightful King.
Other skills worth picking up include Miracle from Cleric, Renewal from War Cleric, and Luna and Dual Guard+ from Great Knight. Although I generally avoid the Archer class Bow Breaker from Bow Knight isn't a bad option to pick up as it can help with her disadvantage at range and her lower defense cap. Hit Rate +20 from Sniper could also be worth considering, but unless you are playing on harder difficulties that extra hit would likely be wasted as chances are you're going to hit everything 100% of the time anyway. Sage has nothing useful for Lucina so unless you just want to collect all the skills or like rally skills it would be best to avoid this class entirely. Lancefaire from Falcon Knight is also something worth considering if you want an extra boost to Strength with lances, but if you're gonna use Parallel Falchion the majority of the time like I do you're likely better off with something else. Discipline from Cavalier is good to have if you want to level her weapon skills faster.
When I have all the skills I want I will reclass Lucina back to Great Lord. By this time all her stats should be maxed out (the exception being perhaps magic, but she really dosen't need it anyway). Because you are only allowed a max of five active skills at any given time you have to decide which skills you want to use. Skills I like to for sure use are Aether, Rightful King, and Galeforce. The last two skill slots I am somewhat torn between Pavise, Aegis, and Renewal. All skills are amazing so it really comes down to what works for you. I guess I haven't completely figured that out yet. Eventually I will.
As an added bonus here is a screenshot of Lucina's stats and skills from one of my playthroughs.