Notes

I'd suggest being careful about it. I once whispered a Blackmailer asking him if anything happened to him the night before. He said no. I told him that X visited him. He was jailed and lied about it. I got Jailor killed.
Usually what I do is, when I see someone visit my target and the target is still alive and doesn't claim attacked or anything, I whisper the person who visited my target and ask them why they visited. Usually, since this confirms me as Lookout, they tend to tell me their role and we team up.
If there were no mafia kills, I'd be careful who I whisper since the attacked person might have been immune (like the Arsonist in your situation).
Lookout is my favorite role and is, by far, the one I'm the best at playing. Your main goal isn't to hide in the shadows like a Spy or collect information about evils like Invest or Sheriff, but to build a team of confirmed Townies. That way, you tighten up the claim space and make it easier for Town to catch evils.

Here are a few tips, note that those tips are personal to me and some people might not agree with me:

When choosing a target to watch, especially in early game, you have two options: choose #1 on the list because Jailors tend to jail #1, by doing so, you'll be able to confirm yourself to Jailor the next day OR choose someone that will most likely get targetted by evils like someone in the middle of the list, someone who spoke D1 (CAREFUL OF VETBAITS IF YOU'RE NOT SURE DON'T GO), or even someone who has a noticeable name. When choosing a target on N2 or N3, pick someone who spoke a lot and was active during the day. You might catch a TP or even evils.
ALWAYS keep track of the visits and use your information properly. For example: you watch X N1 and no one visited. On D2, someone claims Sheriff and say X was Mafia. Be smart, X wasn't visited. The Sheriff is Executioner. Don't be afraid to call him off with proof. Don't forget to do the same with wills, someone posts his will and claims having visited one of your target and didn't show up? He's lying, you just found yourself a fake claim. Your target claimed attacked and healed but you only saw one person visit? They are immune and were not healed.
This is for every TI role in my opinion, but especially for Lookout. DO NOT BE AFRAID TO REVEAL YOURSELF. You have a lead on D2? Share it. Even if people call you Executioner, ask someone else to invest them, ask Jailor or Vigilante to kill them. Show that you do not care if the person you caught gets killed at night.
If town seems to have trouble trusting you, post your entire will, do not just say ''X visited Y last night he is Mafioso.'' Showing that you have collected more information will make you seem more trust-worthy.
You have multiple TP claim? You don't know who to trust? Whisper them and ask them to visit a person of your choice to confirm them. Of course, try and do this if you know you have a Transporter, a Medium, a Retributionist, a Lookout or another TP to have your back in case one of the TP claims decides it's the perfect opportunity to kill you.
Try and build a Town team via whispers, this secludes evil and tighten the claim space. Mafia will also be scared to attack one of you since he has no idea who is protected and who isn't.
Don't be afraid to ask someone why they visited X (try via whispers so you don't piss off a TP or a Jailor into forcing them to reveal). Since you know who they visited, it confirms you as Lookout and any smart player will team up with you.
Pay attention to the available roles. As a Lookout, you collect claims rather than actual information on a person's role (except for who they might have visited). It is mandatory that you master the role list and the role possibilities.
Before you even ask someone for their role, pay attention to what is said. Your target claimed roleblocked? He was visited by an Escort or Consort, no need to ask them their role. Your target is visited by two people and claims attacked in jail? You saw the Jailor and NK or Mafia. This situation can be pretty tricky. I would suggest claiming Lookout and asking Jailor to jail you to tell him who visited. However, not all Jailors will comply and this makes both of you lose a night of extra information which you could have simply shared with Town. If TP is still alive, I will usually go and out them both and hope for the best. However there is a few secrets about the Lookout role.
Here are a few of the role's secrets and mechanics:
When there are multiple visitors and your target dies, the last person who visits is the killer (almost all the time). By knowing this, you will be able to lynch the Mafioso without outting the Jailor. However, not a lot of people knows this so town might not believe you.
There is a bug that causes the Jailor to not appear on your visit list sometimes. It might happen, don't go too crazy and if you get accused of lying tell town about the bug.
If you get controlled by a witch and she makes you watch yourself, you will see everyone who visited you, including the witch.
If you die to Werewolf watching someone who also died to Werewolf (or was healed from the Werewolf's attack), you saw the Werewolf, stay around to tell the Medium or to see if you get revived.
If Escort or Jailor roleblocks or jails the Werewolf or Serial Killer, you WILL not see the killer visit them, even if you watch them. Meaning the person that THEY visited, is confirmed to be the killer.
You will not see the Survivor put a vest on. Or Veteran alerting (I mean... you die).
You will see the BG and the Doc use their vest or self-heal and you will see the Arsonist visiting himself when igniting.
If the witch forces someone who usually doesn't visit (or can't visit on certain nights) to visit your target, you will see them.
You will see the witch visit the target they control, but you won't see the witch visit the one they send their target to.
You will see the Transporter visits himself when he transports himself.
You can see a Medium seance.
This last one I am not sure, but I think you can see a Jester's haunt.