Hey, welcome to the Forum!!
Ahh.. the Angeles National Forest, I've romped a lot of miles there. Have you ever been Malibu Creek State Park?
They were building a camp site there when I left CA, you can only camp in the campground there, but there is some great hiking there. I've hiked the creek from the lake to the lagoon and that's a lot of fun.
This time of year (back in the `80s) I used to go swimming in the swimming hole below the dam for the lake, and on weekdays I had it mostly to myself. Don't know if they restrict that now, but that's a great swimming hole in the early Spring.
Others have given you so much great advice already I'll comment more on the trips you might consider...
The Angeles NF is really too hot and dry in the summer to go backpacking. Water is as rare as gold. I knew a few spots where it might be found when I roamed there, but they are all rare and inconsistent sources and the further you get into the summer the less likely you are to find any water. The only tip I can offer on that is the further you hike up the bottoms of the canyons the more likely you are to find water, and the less likely you will be found if you get stuck there.
However, the Angeles NF has some awesome spots waiting for those willing to explore a bit from late Fall to early Spring, and they can be like little "Shangri-Las" when you find them.
I used to hike Piru Creek a lot too. I'd park below the Pyramid Lake dam off of I-5 and bushwhack downstream. I never spent the night in there, but I don't think they were any regulations against it. In the winter they'd stock the creek with trout and it was pretty nice in there. I never saw a soul once I was 10-15 minutes into those hikes and it always amazed me that I had it all to myself.
There were also some spots I'd go behind Castiac Lake, and while I mostly car camped back there, there were some great spots to spend a night or two backpacking there in the cool season.
If I recall correctly, Early Fall might be a bit too late for backpacking the Sequoias. I recall the backpacking season runs from mid-July thru mid-September up there. Before and after that you still have a good chance of getting hammered with deep snow. But it's sweet in the Summer, so you might want to go there sooner.
There's really no reason you should wait to go there, you can test your gear and shake out the bugs and get some experience there, just plan on staying close to your car so you can bail out easily if you want to.
I used to drive up I-5 to Bakersfield, then to Porterville, then to Springville, and enter the Sequoia NF from there. There were lot's places to car camp there, and I'd park and either car camp or head into forest and camp out. I didn't use a tent much back then, I just slept on the ground in my bag. I don't know what the rules and regulations are there now, but no one ever bothered me or my car when I did that, and there was no charge to use the campsites I went to, which were mostly only used by locals for hunting trips, not those used by tourists.
You should definitely check out some of the Trip Reports here by the member "Wondering_Daisy". She's got a spot or two she hikes along the coast up north of you and the photos she's posted are stunning. I'd love to do those hikes and they might be perfect for you to get started with. And she's posted reports on the kinds of places you are most certainly dreaming about going as well, after you have some experience, and you'll learn a lot if you take the time to read them.
As for gear, I'll add these tidbits:
I mostly use a "Super Cat" alcohol stove now. They work great for freeze dried "Mountain House" type meals and instant oatmeal, coffee, tea, hot chocolate, etc. Not so good for frying bacon or grilling steaks. Depending on the food you bring, if you're only doing 1-4 nighters they are a very good choice.
I have several "Multi-Tools", all cheap knock offs of a Leatherman, but I don't ever bring them backpacking. I do carry a very small pocket knife called a "Cowboy Toothpick". It's light, handy, and all I've really needed. They cost about $3-$5 bucks depending where you buy one.
"You want to go where?"