Training weapons are great since their weight is very low, they are pretty accurate and have great durability to boot. If your character has very high Strenght but low Speed, they may actually be the better option (that, and for Monk characters that can achieve x4 attacks that way), if only to avoid getting doubled.
Iron weapons are the bread and butter, thanks to higher Might. Slightly heavier than Training ones, but not that much in most cases.
Steel weapons should be used only to get the extra boost necessary to kill a target, or against units with high Defense, since their weight will hinder one's speed too much and make them susceptible to getting doubled (and eventually killed). Unless your character has insane Speed to begin with, that is, or Speed so low that using lighter weapons would be a waste anyway (ie. armored units).
Silver weapons are better than everything listed above, thanks to very high Might, similar accuracy to Iron (ie. pretty good), and being lighter than Steel weapons (still a bit heavier than Iron ones). The caveat is that they have low-ish durability, and a B rank is necessary to use them.
TLDR: Training and Iron are great in most circumstances (using one or the other will essentially depend on your character's speed).
Steel are situational. Silver are amazing (although can be considered overkill in some situations).
