How can I start playing Stellaris ver. 2.0?

I think after about 1000 hours you can say you finished the tutorial

it not deepest of the paradox game the deepest will be Victoria 2 personally
stellar come to control space is more a 4 X you must explore first and control your solar system and those surrounding your explore
all planet and asteroid and build to collect that extra: energy, mineral, and science that will speed up your grow out of this phase after that will need to expand as fast as possible with your planet limit you may need to make sector ( you will need to build the sector to optimise them the ai is brain dead there) phase 3 you will need to build a decent navy to protect your border and maybe expend a little on easy prey while your build your economy with planet building and recherche resource building upgrade you will need a strong economy to fight off any crise or any future war you will wage( better be ready than scred) phase 4 conquer or invited to your federation ( if there is no criss happening) if there is a crise you need to deal with it they only get worst with time

It's actually easier to learn the game since 2.0 with the introduction of starbases, FTL and fleet reworks. And I'd say (after spending thousands of hours playing other Paradox games) you don't really need to play them to start playing Stellaris. But getting to the point, things that you should consider when playing first few games:

1. Default setup/difficulty settings can be too hard for new players (especially if you haven't played strategies a lot), don't be shy to to lower number of AI empires, disable advanced neighbours, nomads or even fallen empires - it'll help you to learn the "pacifist aspects" of the game like exploration, colonization, economy, etc.

2. Play vanilla game mode (without mods), or only with cosmetic ones. Mods are awesome but they usually change mechanics of the game too much which will confuse you if you later play a normal game or multiplayer without mods

3. Start first game(s) with normal government ethics like xenophile or spiritualist. Playing as a mind hive or machine intelligence is awesome but the gameplay mechanics require some experience or reading a lot in the wiki

4. If possible, play multiplayer with a friend (whether he played Stellaris for longer than you or not). Having a reliable ally helps minimising effect of newbie mistakes that you will make and can help with the very bad events called "crises" which have to be resolved by force.

Is playing the base game fine or is buying the DLC(s) suggested?

Just start in Ironman mode and treat it as a roguelike - helped me alot in finding maximum enjoyment.

The amount of savescumming i´ve saved myself from is insane!

Just try and try. Each try is different but it's better to make more differences by adding knowledge of new mechanics from guides or own experiments.