This is just my feedback about what it was like to take my budgie to the vet. I'd been putting it off as I was really worried what it would be like, so I want to reassure other new budgie owners about the experience.
My little Calcifer had diarrhea and (I think) a bit of a temperature but no other symptoms. I'd been putting off taking him to the vet as I was scared what the news would be and I thought the journey would be very stressful for him.
What I ended up doing was to get a small cage and pop him into that, and cover the cage with a cloth. He was fine in the car, even took a little snack of seed as we were driving!
The vet took him out of the cage and examined him very thoroughly, looking at his vent and sort of palpitating him. He also listened to him through the stethoscope, which was funny as the stethoscope was bigger than the bird
The vet asked me lots of questions about Calcifer's behavior and symptoms. Finally he said that most likely, Calcifer had a bacterial infection brought on from being stressed by his changed circumstances (he's a new arrival and still a baby) and he's probably also stressed from being alone in quarantine and can sometimes hear my other budgies.
He gave me some liquid antibiotic to put in Calcifer's water bowl for the next 5 days. He also complimented Calcifer for being very well behaved and tame - I'd been working very hard on making him used to being handled so I was very pleased by that!
Calcifer's diarrhea cleared up after only a day. I've still got one more day of antibiotic to give him, but I'm hoping he's over this. I also cleaned his cage and all his toys very thoroughly with cider vinegar in case there's still some bacteria lurking about to reinfect him.
Anyway, I just wanted to say, if you DO think it's necessary to take your little bird to the vet, just do it. It's not as stressful as you think it will be and hopefully, the news won't be bad either.