Rundown on Yerushalayim public transportation:
There are bus cards called Rav Kavs- they are green and purple and will get you on any public bus in the country (while you may need to pay extra for different locations or distances on intercity buses, they can be loaded onto this Rav Kav). They can be purchased on buses or at the Tachana Merkazit. You can load money on them either on the bus or at machines at light rail stations. (I don’t remember ticket prices but it’s cheaper than NYC and you have free transfers for 1.5 hours.)
There are many local buses in the Geula/Meah Shearim area, as well as a bus station that has buses to probably 95% of the cities in Yerushalayim you’d probably be interested in visiting. There are two area buses, the 1 and the 3, which will take you directly to the kosel, but like people said they do go through an Arab neighborhood for a few minutes. 99.9999% of the time there is no problem, but I definitely understand preferring not to take them (especially as I was among the .00001% of people who did once have a problem…). It is also a quick walk from Meah Shearim to the kosel, but you will probably want really good directions or someone to show you around.
You’re also a quick walk from Meah Shearim to the center of town, including Machane Yehuda. Many stands have great hashgachas, including Eida Chareidis.
If you’re there for Sukkos, I also recommend going for Birkas Kohanim if only so you can just see the huge numbers of klal Yisrael all in the same place.
There are kosher beaches in Ashdod, Netanya and I’m sure other places as well.
There are several buses to Beit Shemesh, including the 415, 417, 418 and 420. They all go to roughly the same places but have different routes, so check with your hosts for the bus you should take. You will have to buy a ticket on the bus (though you can buy a return ticket to be put on your Rav Kav). If you want to go to Ramat Beit Shemesh, take the 417 specifically.
A good bus tour can be a great way to get a lot of sights in that you otherwise would have to get to yourself. Basically, sightseeing in Israel is either cheap or convenient. I’ve done both routes- the pay-for-a-bus-and-get-ferried-from-kever-to-kever route and the so-the-bus-to-the-nearest-town-comes-in-an-hour route- and for a tourist, a bus is definitely worth springing for. You can get basically anywhere by public bus (including Yam Hamelach like you mentioned) if you really want to, but you should probably look into more practical methods.
When in the Old City, don’t freak out but also stick to main areas unless you know where you’re going. While people there are really nice and mostly speak English, if you’re not familiar a lot of the streets look the same and can be really confusing.
Have an incredible time!