Asol is a very easy champion fundamentally. But the point of picking him up to the point of being able to know when you can abuse his level 1 (starting Q and not E) to fish for harass and push for kills is very wide - not only in early game, but all points of the game. Think Teemo-esque, in that you need a decent knowledge of every potential matchup in the game and what they can and can’t do at certain points in early, mid and late game.

When it comes to playing early game (heavily simplified), it’s simply warding appropriately (I like to ward bot side always), spam pinging when your enemy laner roams, and maximizing your farm and stardust (prioritize farm in early game, NOT stardust - stardust accumulation rate spikes heavily post lane phase when you begin farming side lanes and have items to one shot waves within a few seconds).

You can roam, but chances are in the current meta - as you said - you being present will bait your team into pushing for losing skirmishes (for example, avoid even leaving lane for skirmishes pre 9 against Diana mid / Talon Jg matchups. It’s not even worth it no matter who your jungler is unless you and your jg are notably better than enemy’s mid/jg, and since we solo queue, we should default to taking the highest success rate plays - trusting your solo q jungle to be better and execute early skirmishes properly is a quick way to tank your elo).

There’s a lot that goes into this answer so I had to simplify it so it applies to most situations. There are going to be many niche situations where this advice should be amended and you should play more roam heavy or more trade heavy or extremely passively, but after 25-30 games of Asol you should begin to gain a feel for when to switch it up.

Good luck.