Is she under 18 or over 18? That makes a difference in how the police respond.
Really, that aside, there's enough variation that you could write it so you are justified in the police doing a bunch of different things, and all of them would be "realistic."
You want an "all out response," make her under 18, and a possible stranger abduction. We're talking Amber alert, media coverage, the works. Even a non-custodial parental abduction is usually taken pretty seriously these days.
If you want to write it like the police "don't care," make her a troubled youth with previous police interactions and have her possibly have run away in the past. There'd still be a report, she'd be entered in the system, etc, but there wouldn't be that same sense of urgency.
As to the boyfriend, you could justify a bunch of different options there as well. Everything from finding out where he works and interviewing him there, to calling him to ask to come in and talk about it, to visiting his house unannounced and asking him about her.
The more apperant urgency in her disappearnce, the quicker and more "intrusive" (for lack of a better word) the response.
They shouldn't be able to arrest him without some probable cause he's involved and they shouldn't be able to do a non-consental search of his house without a warrant.
The thing is the police would very likely ask if they could "look around" to make sure she's not there and tell him things like "We just want to see if she's here. We don't care about anything else" (which may or may not be true) or "You can say, "No," but that looks suspicious", etc, etc.
Many people will consent to such a voluntary search, for varying reasons. Some want to honestly cooperate, some feel pressured, and some people just don't want to "look guilty."
Don't forget that police are allowed to lie to interview subjects and suspects in an investigation. (They can't say things like, "We have a warrant to search," when they really don't, but they can lie about what they are looking for and whether or not they really will ignore the pot on the kitchen table when they go through the house looking for her, etc).
EDIT: I missed the "last seen at boyfriends house" the first time. In that case, yeah, I think they'd show up there, unannounced, to ask him questions and ask for a consent search. Depending on how that goes, and what else they know, they may also try to get a search warrant for his house. (Which they may or may not get, depending on the circumstances)
But, if that's the last place she was seen, they'd definitaly want to talk to him, get his consent for a search, and see where that leads. If he declines to give consent to a search, or seems "evasive" that would trigger more interest in him. (Which could still be wrong. He may have nothing to do with it, but doesn't want cops to know he's growing MJ in the basement, etc. )
Having the police get suspicious of the boyfriend because he's being evasive for some other reason could be a plot complication. They could waste time following up on him, get a warrant, etc, only to find he has nothing to do with it, and they shoudl have been lookign elsewhere.