How do I go about this....?

1) To an outsider, this could look suspicious. Two showers? I can understand before and maybe after work if you're job is physically demanding or you're going out with the girl after work, but if he's running in and taking a shower RIGHT away, and then doing it right before bed, it looks suspicious, as if there's something he's either trying to hide or that he's getting sweaty and rank partying, which seems a bit odd. If you trust him, then I'm not going to be butting in there.

2) There are several ways to do this. I would only suggest the "nagging" (you may not feel you are, but he may feel differently if you come across as accusing him of something or pointing out a fault."Why don't YOU ever take ME out?" kind of questioning should be a last resort. And telling him that you may not be there when he gets back may sound like you're angry, and threatening him. Be cautious about how you handle this one. Nonchalant. "Hey, since you're going out, that sounds fun. I might hang out with some of the girls tonight. I've been stressed all week, and I haven't seen them in a long time."

You want to come across as a person who wants some fun. You want to be in his life.

Does he ever talk to you about his friends? They seem cool, right?

You'd like to meet his friends. Or if you've met them, but only once or so, "Hey, have you seen _____ and his girlfriend lately? They seemed cool. We should double date!"

As a part of THIS idea, you could meet them and set up some time to hang out, some time where he has a lot of notice about doing it so he can schedule properly, and you make it something fun. Perhaps if you know someone he hasn't hung out with in a while, you suggest he get together with this person, and you come and meet them too. If it's a friend he hasn't seen in a while, he might be more enthusiastic about the double date idea.

Also, you could host a party. If it's around his birthday, hold a "surprise" party for him. Invite friends over to your place for the party. You get to see them, meet people, and he gets a party; and you're the sweet girlfriend who set up this big party for him. This might require a little sneakiness on your part however, so if you can't find his friends information, if he's super cautious about what he lays down, then there are other ways.
You can also get him to invite some of his friends over for "the big game." or some movie marathon he would like. You're the cool girlfriend who lets him watch sports with his friends, and you get to meet his friends, and maybe show some interest in what he's doing. You win.

Do this a couple of times, with patience. If he shoots you down repeatedly, or "forbids" you from joining him, you can either leave or stay, depending if you feel this is acceptable or not.

I wish you the best luck in your decision on how to approach this.

If he's a boyfriend, of course he should ask - or it should be assumed you can go. For whatever reason, he seems to think that what he's doing is OK. Not only is it not OK, but it makes me wonder about what goes on when he's gone. You could approach it a couple of ways. One is just ask why he never invites you; Another is when he says he's going, just say, "hold on. I'll grab my purse and go with you." Then see how he reacts. If he shoots that idea down, then say "OK , if you're looking for me, I'll be out. It may be late, so don't wait up." Or if you don't want to try that, just have a few of your own friends hanging out when he gets back. But I have a gut feeling based on how he is acting, that you're going to get opposition to any path you take here. Or maybe indifference, but not quite as likely. Something is very much amiss here.

ok. well I say you cut get out of this relationship. he is not worth your time. if he doesn't invite you to go with him . then he is probably hiding something or he hasn't told no1 about you. honestly I think he is CHEATING ON YOU. a REAL BF. shouldn't do that. he would ask his Girlfriend if he wants to come. if you don't want to let him go. then get some CONFIDENCE TO ASK HIM WHAT IS HIS PROBLEM. and WHAT HE WANTS WITH You. communicate. and ask him. don't be afraid to ask . don't be afraid to tell him how you feel or what you think. if he doesn't give you a reasonable answer then its best you MOVE ON. I'm SORRY. BUT I THINK YOU SHOULD JUST MOVE ON. I been there I know HOW IT FEELS. well hope I could help. bye.

Well sweetie - I have seen it all - I'll bet if you do a little digging or a little looking around, start just going with him - you'll find most of his friends will be shocked he even has a girlfriend - why isn't he inviting you with? It isn't about trust - it is about chemistry - and he doesn't have it for you if he isn't willing to spend any time with you.

Cut your losses & move out - he probably won't even notice you are gone. Sorry to be the one to tell you, but he isn't interested in a relationship with you. Additionally, he is probably seeing other people or worse many other people.