I'm making a new character that wants to be a war magic or bladesinger wizard, I'm not sure which subclass to choose and what I should multiclass into, my goal is to make a tanky spellcaster that is great in combat and out of combat.

The wizard is a fantastic class, you will have things to do inside and especialy outside combat and a great character whatever the subclass without problem. However there is a gap in power between these 2 subclass.
Bladesinger is one of the strongest wizard, and the most resistant one just necause they can use armor. It can use blade song to be able to do a lot of damage with booming blade and multi-attack, or just defensivly to boost his concentration. It can stay at range with other cantrip (like toll the dead or firebolt) or in melee with weapon. With the war caster feat it has very little chance to loose concentration on spells which is insane. The only obligation is to take green flame blade or booming blade as one of your cantrip to use its full potential (I like the idea of firebolt+green flame blade with elemental adept for one of my futur character)

In comparaison war caster get one free buff for damage every day, an a defensive feature not as good as just casting shield or silvery barbs. It is a weird mixt between abjuration and evocation wizard and worst than both.

If you want defensive option I also suggest the abjuration wizard or the order of the scribe, they have great defensive feature. The abjuration wizard is all about protection and the scribe is the most versatile wizard with a feature to protect himself also