Adam Forshaw curled home the Bees' opener with 35 minutes on the clock and after Blades defender Neil Collinshad levelled things up from a corner, Brentford took control with striker Will Grigg grabbing a brace and were good value for their win.
Former Walsall striker Grigg missed the chance to grab a hat-trick when his late penalty was saved by George Long late on.
Weir said: "Rather than saying what we're going to do and what we're going to get, I just want to improve as a team.
"There are aspects after this we can improve on and that's where we'll start on Monday morning.
"I'd much rather concentrate on the short term and get the balance between everything right before saying what we're going to achieve this season."
Weir does not see his relative inexperience as a manger being a hindrance to his side's development and believes the entire side will learn from their mistakes.
"It doesn't matter how long you've been a manager, you're always learning and I learned here," Weir added.
"There were a few positives and negative things to come out of it.
"We did well to come back into the game, but when you come here and lose three goals, it will always be difficult to win."
- Sports & Recreation