We have an old lilac tree which is putting out some strange new branches. They are VERY straight. Perhaps I think they're strange because the rest of the tree is so "gnarly". Is this normal or are they some kind of sucker?
Sounds like their root suckers, Horseshoes.
These probably wont flower for some years and are best removed to retain the original shape and charecture of the shrub.
If you can get in there with a pair of secateurs or loppers and cut them off at ground level.
Ah, if they are Not root suckers, they are probably what are termed as unproductive branches and are best pruned out after flowering (usually late May)
Good Luck
Chip