The Veronica is a perennial. Cut back the dead stuff; new growth will come up from the ground.

Butterfly weed--are you talking about butterfly weed (Asclepias tuberosa) an orange flowered perennial or Butterfly bush (Buddleia davidii), a woody perennial?

Weigelia--I don't grow it but I believe it's usually pruned pretty hard after bloom.

Not familiar with the fourth plant.

weigela NEVER needs to be pruned... but some can get as big as your house.. so they are pruned for size ...

ALL flowering bushes are pruned AFTER flowering.. or you wont have many flowers if you cut off the buds ...

pruning shrubs involves removal of up to 1/3 of the largest branches as close to the ground as possible ... research this when the time comes...

same logic with burning bush.. if its the same type i have ...

for a newbie... it should be rather easy ... wait until your plant just start budding in spring.. and cut out all dead stuff once you can ID such ... there never is a real hurry with shrubs/bushes/trees ....

you can always take it off later... you cant put them back on ...

good luck
ken

From Tracy DiSabato's book The Well-Tended Perennial Garden (this book is a must-have):

Butterfly weed (asclepias): leave up for winter, cut back in spring. The same is true of butterly bush, the enemy of both is winter wet and leaving them up helps to withstand this. No info on making asclepias bushier, but says deadheading results in rebloom about a month after first flowering.
Veronica: Can be pruned down for winter. To promote rebloom, you can shear after first bloom - keep moist thereafter. Sunny Blue Border requires frequent division (after second or third year) to maintain strength of plant.

Its astonishing what a little maintenance will do to make your perennials look like a Horticulture photo shoot!!