We know the speaker is the one going to the wars, but here it could also mean that Lucasta is the one going to the wars! This suggests that even though she will remain at home, in many ways she, too, is suffering from the effects of war because her lover is leaving her.
To war and arms I fly. (4)
The action of going to war—to "fly" (which would have been pronounced as "flee")—rhymes here with "nunnery," which suggests some type of strange connection between the chastity and holiness of a nunnery and the act of going to war. It's tough to suss out what that connection might be, but perhaps it has something do with the way in which the speaker views his decision to go to war as something driven by "honour" (12).
True, a new mistress now I chase, The first foe in the field; (5-6)
The connection between war and love is again suggested in these lines, where the speaker refers to his enemy ("foe") as a potential "mistress." What is interesting is the hint at cheating, in that the speaker is abandoning one mistress for another. This perhaps implies something shady about war, in a poem that wants to justify the decision to fight for honor.
And with a stronger faith embrace A sword, a horse, a shield. (7-8)
Once again war and love are intimately linked; the fact that the speaker "embraces" the weapons of war reminds us of the loving embrace associated with a romantic relationship. That he "embraces" his weapons, however, with a "stronger faith" is more than a little problematic here. Why is it stronger than the faith with which he embraces his main squeeze? We're betting she's not gonna take this well.