Feel free to follow suit
While it can be tempting to formalize all exchanges if it's what you're accustomed to or because it's how you were taught, a lot of the time, it's just not necessary. Pay attention to your workplace cues, or you'll just end up sticking out. If your boss forwards you an email with nothing more than a note about taking a look to see if it's of interest, and you reply with a formal message, you're not winning any brownie points — you're only clogging up his/her inbox and ignoring how your team handles casual correspondence. Or, if your colleagues start communicating with you in a casual way (without addressing you by name or including an official sign-off), accept that as your sign to respond in kind.
Switch it up (depending on who you're talking to)
It doesn't matter if your last boss made it clear that adopting an informal tone was intolerable; you're not working for his/her anymore. Though, a word of caution, just because you reach a point where you drop the "best" with your manager doesn't mean you should abolish the word from your vocabulary altogether. If you're regularly speaking with someone you've never met and your relationship is more formal than not, don't be so quick to sign off without including a proper closing, especially if you're on the fence about how to proceed. Erring on the side of caution will always be sound advice.
Consider timing
If you go on vacation for a week and return to the office with a list of questions for your boss, who you haven't spoken to or seen since you went away, it's probably best to begin that first email back with a pleasantry such as "Good Morning," "Hope you've been well," or, if your manager was the one on holiday, "Welcome back" — even if you and your supervisor usually skip the greeting or small talk. Minding time lapses is also important for exchanges with outside clients or vendors. If you're in touch once a week or just a couple times a month, it may be appropriate to start off the initial message after some time has passed with the go-to intro and the appropriate closing signature.