The problem unlike with animals is actually getting the drugs used for the lethal injection. For most veterinary uses they use something like Euthanal (which is the barbiturate pentobarbitone) and this causes rapid deep sedation and respiratory depression almost immediately after intravenous injection, which very quickly leads to death. Quite high doses are needed though in comparison to invoking just sedation (in case anyone is wondering why I know this sort of stuff, I did a Pharmacology degree recently which covered the actions, side effects etc of most major drug classes).
The main problem is that most manufacturers of pentobarbitone, sodium thiopental and similar drugs will not allow their supply for use in executions which is why their having to use various different protocols to try to get around the supply shortages - in theory it doesn't really matter as any anaesthetic agent will eventually lead to death in overdose, but as to the length of time etc it's debatable as you obviously can't test it on humans, and I seriously doubt that any ethical review board would pass an in-vivo experiment on primates or similar.
Traditionally a 3 drug cocktail of a barbiturate, a muscle relaxant and potassium chloride are used, mainly to ensure a "painless" and rapid death as all three of them cause symptoms which will lead to death (by sedation/respiratory depression, respiratory depression [by paralysis of the chest muscles - plus the agent also stops involuntary spasms prior to death] and stopping the heart respectively). Some states have been switching over to using a single barbiturate or benzodiazepine type agent, but then there is the danger of the agent not working satisfactorily due to tolerance etc (which there is a much greater chance of in those who may have abused drugs in the past), which is part of the reason for the uproar that's currently going on as some of the newer protocols don't perhaps work quite as planned.
As to whether I support the death penalty or not, to be honest I'm not sure. Part of me says "hang em' high!" for the most dangerous, sadistic criminals, but the other part says that imprisonment is more of a punishment and there's always the danger of miscarriages of justice (plus the expense of the government being possibly being sued over executions).