There is no easy answer to this. It depends on who is on your team and who is opposing your.

Generally speaking Lili has a lot of healing output, but won't have a lot of damage, unless she throws a lot of talents at that - wich generally is not such a hot idea. So on average you belong to where team fights happen. If the lanes are somewhat even and no team fights are being forced, she tends to be best going with someone who can pressure but is squishy on his own early game like Valla.

If You aren't forced to leave your lane due to team fights or objectives it is very benefical to have at least one person in every lane - at least to level 10, so if no one is there you might want to stay for a while, but it isn't great.

Being the sole survivor depends on who you are. As Lili you can annoy in early game if there isn't like their whole team or too much burst, but generally - run. There isn't much Lili can do to stop/slow down a whole team, like Zagaras Maw or Zeratuls Void thing. Even when you are specced for damage you won't have burst like Nova or Jaina, and you have no hard escapes to get out after some pokes like say Tassadar. So while you can poke/annoy a bit before 10, you will get stuned/murdered fast on your own later.

Not to go too much into stuff you will learn youself later anyway, but the whole team fight/dying/last survivor thing naturally changes with the course of the game, as you could risk more early game where you are back in say 10 seconds or 20, but if you die and are out for a minute while being your teams main sustain as a lili healer, the guys spawning before you who died during the fight will end up being useless for the most part.

Cheesy or not, my best tips for HoTS are try to have fun, throw people who are toxic on ignore and don't let it get to you. Everyone has a bad run here and there, some games aren't meant to be won, and you aren't bad just because things go bad here and there so fuck people who try to make their team players feel that way.

Other than this, if you have the time read a guide here and there, watch some videos (streams or youtube - personally I like to watch pallytime on youtube every once in a while since he is a positive guy) and try out different heroes. It is genreally a good idea to see what all heroes can do, experience some of their strenghts and weaknesses first hand. Level 5 is a fine goal, gives you all their talents and bonus gold.