(Sorry I kinda alternate between second and first person, between recounting experience of what I do and giving advice, may make for a weird read so apologies in advance)

For me house party or bar it's usually:

• arrive roughly half an hour to an hour late. (If it's a close friend's party however I'll arrive early or on time to help set up and kick the party off right, decent friend maybe 15 mins late)
• try to find the host and say hi, if they're occupied talking to someone else, then no worries, give them a wave and hopefully circle back at a better time (if the host knows you've arrived, and they're a good host, they may help introduce you to people later e.g. they say to whoever they're currently talking to "Oh my mate luklmy is here, he also has a beagle, I'll introduce you!". This gives them a chance to catch up with you, introduce you to someone new, and then excuse themselves to work the room
• a good opener is "Hey I'm YYYY, how do you know [host's name]?" it gives you a bit of an insight into the person (are they a school mate? Diving buddy? Valorant squad member?) and your first springboard into a conversation. If they don't have a good thread to pull at, then it'll turn to you to share how you know the host. It can be good to roll into anecdotes from here e.g. "[Host name] was such a brat at school, did she tell you the time that..." or "[Host name] is a beast at Valorant but she wouldn't let on, last night it was 3v1...". That'll give you the threads to continue
• it can also be good, when appropriate, to ask "Hey do you know anyone else at this party?" and give them a chance to point out more people, or call them over into the conversation, now rather than a one on one convo, you'll have a bit of a group convo going on. You may be involved (great! You're meeting more people) or you may get kind of closed out as they talk about in group stuff. If the latter then no worries, you can make a clean exit by saying "Hey it was nice talking to you all, I'm gunna go say hi to those people over there" you could even double back to see if the host is free
• if a conversation is going well and we're vibing, I may say something like "Hey can I get you a drink? Do you know where the esky/fridge is?" or "I'll go grab us some chips. Which way is the kitchen?" or whatever then make my way over. With any luck there may be someone at the esky who I can strike up a convo with, and if they also seem cool then say "Hey come over and meet [person you were just talking to]". The person you were just talking too as well may have started talking to someone new in the time you were going to get a drink, so you've got another person to meet straight when you come back! Too easy!
• if a conversation is getting stale or going poorly you can break it off in a few ways. Similar to the above you can say "Hey it's been really nice talking to you, I'm gunna go say hi to person over there/get myself a refill/check up on the barbecue/dip my toes in the pool/find the bathroom" then go do that thing (you don't want to be the person who ends a conversation, then walks around the corner to stare at their phone, then the person you were talking to comes around and sees you.
• If you smoke, you can take a smoke break. Honestly, this is terrible advice, but I was a social smoker because it gave me a ready excuse to break from a noisy party. You've also got a ready made group of people to meet. Even if you don't smoke, you can go hang out with them with the line "Hey I'm just gunna go get some fresh air" then make your way outside.

There's a bit of ebb and flow at the start of the party as the vibes settle in. As you probably noticed there's a lot of making polite excuses to move around the party or draw more people into your convo if you prefer that. Once convos get big enough and people have put away a few drinks you may get people suggesting drinking games or Smash Bros or a dip in the pool or whatever it may be. It's fun to get involved.

For me, if I'm drinking, I may hit a point of being buzzed enough that it feels like second nature, I become a lot less self conscious, and enough stuff is happening around me to react to that things don't require effort to get going.

If however I've had a few drinks and things don't take off, and everything still feels stilted despite my best efforts, I'll probably just call it a night. I'll find the host if I can and make an excuse (Have to feed the cat/last train home/visiting my parents tomorrow) if it's a good friend I may say as much "Hey man just not feeling it this evening, gunna make an early mark, thanks for throwing the party". On the way out, if there was anyone I particularly enjoyed talking to, I'll make sure to say goodbye and if I haven't already I'll ask for their contact.

As dasharezone says, if you're not feeling it you can just hit the bricks.

Hope this makes sense, it's kind of a base format that is essential to most kinds of standing party, alcohol or none, daytime or night.