Traveling and living in wormhole space (also known as w-space or j-space) is somewhat different than traveling in known space (k-space) or highsec, lowsec and nullsec. For example the only wormhole with NPC stations is Thera (get the location for Thera from here), but all other wormholes are either uninhabited or have player owned structures. There are some wormhole freeports that are open to everyone though.
There are also no stargates that connect wormhole systems, only other wormholes that must be scanned down with scanner probes. This is why it is important for most ships in wormholes to have a probe scanner, because if the wormhole closes behind them, they will be trapped without it. There is however a rescue service provided by the EvE-Scout corporation for those who get stuck in a wormhole.
Another difference between k-space and w-space is that there's no local chat list. You only appear in local if you type a message there, and then your name disappears after a while. But basically in most cases you don't know for sure who is in a wormhole with you, as ships can fit cloaking devices to stay hidden from the overview and directional scanner.
As you may already be aware, the green anomalies in your probe scanner are sites that you don't need to scan down with probes. These are typically either combat sites or ore sites, although shattered wormholes have ice anomalies as well.
The red signatures you need to scan down, and then they become green or warpable. These include wormholes, relic and data sites, gas sites and now also Deepflow Rift sites, which is kind of like internet space fishing. Red signatures can be scanned down with combat probes, but core scanner probes have higher strength. Sisters probes have better scan strength than regular probes, but they are also somewhat more expensive, although they don't break the bank either.
When scanning down red signatures, you should add them to your bookmarks or otherwise you will need to scan them down again if you get disconnected or log out of the game. Just right click and select Save Location. I recommend letting the bookmarks expire after 2 days in most cases, because for example wormholes disappear after between 16-24 hours, although some rare wormholes can last up to 48 hours. Important note: ALWAYS bookmark the wormhole on the other side, after you jump through it. This is a common mistake and forces you to either re-scan the wormhole or you get trapped.
It is also advisable to bookmark wormholes from your overview rather than from the probe scanner. Because when bookmarking from the probe scanner, you will land where the signature is, which is typically 10km from the actual wormhole. But if you bookmark from the overview, then you will land directly on the wormhole. The exception here is when you deliberately close the wormhole by collapsing the wormhole connections. Every wormhole has one or two static connections to other wormholes, but if nobody scans them down and warps to them, they will remain closed.
Before you do sites in a wormhole, you should do a risk analysis. This involves doing several things, first of all fly around the wormhole using the directional scanner. Check if some corporation is living in there, and if you can see other ships in space. If you can find structures such as citadels or active control towers (check for force fields), that's a sign the wormhole is inhabited. But not all is lost, you can check their Zkillboard profile to see how active they are in terms of PVP. You can also check when they are active, and if it is during the same time you are active, the risk of doing sites just went up.
Another thing to do is check the Zkillboard profile of the wormhole system itself. You can just copy and paste the J-code of the wormhole into Zkillboard to see recent activities in the wormhole. If there has been ships destroyed within the last 24 hours, there is a chance that the wormhole is camped or it is in the wormhole chain of an active PVP corporation. In this case the risk of doing sites went up. Another good site for intel gathering in w-space is Anoik.is, there you can see a lot of data about wormhole systems.
Then you should also scan down all the red signatures and if there are any wormholes, check them out to see if they are inhabited. This is especially important if you intend to do combat sites, gas harvesting or mining in the wormhole. Hacking sites can be completed quickly, but other sites require more time to finish, which increases the risk of being caught by PVP pilots. Just do the same kind of risk analysis in the connected wormholes as you did in the first wormhole, except you don't necessarily need to scan every red signature down.
When doing sites in a wormhole the most important thing is to always keep the directional scanner open and visible. You can detach it from the map by pressing the square icon on the top right. If you are in a green site that doesn't require scanning, then it is even more critical that you keep refreshing the directional scanner to see other ships in the system. I would say do it once every 5-10 seconds if you are in a green site, less frequently if you are in a red signature. You can press the V key on your keyboard to do it as well.
If you see ships or probes on the directional scanner, whether combat or core scanner probes, then it is time to get safe. Generally you should call back your drones if you have any, and warp off to the wormhole from where you came. Alternatively, before doing the site, you can make safe spots inside the wormhole and press Control-L to open up the bookmarks that are in system. This makes it easier and faster to warp to a safe spot. You can make safe spots by creating a bookmark while warping through space.
When doing combat sites in wormhole space, it is also advisable to have a Mobile Tractor Unit with you to gather the wrecks when you do the site. This makes clearing the site a lot faster. If you want to do salvaging, I recommend bookmarking the MTU for every site, and once you're done then you can return with a salvaging ship, like a destroyer, to clear the wrecks. Wrecks stay in space for about 30 minutes.