Short answer: Yeah.
Long answer: it really is both, praise God! For one of the clearest passages on this, let’s look at Philippians 2. In this chapter, Paul exhorts the Philippian Christians to humble themselves and count each other as more significant than oneself. And as the strongest justification for this admonition, he cites what the Savior did, in a hymn:
5 Have this mind among yourselves, which is yours in Christ Jesus,
6 who, though he was in the form of God, did not count equality with God a thing to be grasped,
7 but emptied himself, by taking the form of a servant, being born in the likeness of men.
8 And being found in human form, he humbled himself by becoming obedient to the point of death, even death on a cross.
9 Therefore God has highly exalted him and bestowed on him the name that is above every name,
10 so that at the name of Jesus every knee should bow, in heaven and on earth and under the earth,
11 and every tongue confess that Jesus Christ is Lord, to the glory of God the Father. – Philippians 2:5-11
And to that, we say amen and amen.
Immediately after the hymn, Paul continues with what, therefore, the Philippians should do:
12 Therefore, my beloved, as you have always obeyed, so now, not only as in my presence but much more in my absence, work out your own salvation with fear and trembling,
13 for it is God who works in you, both to will and to work for his good pleasure. – Philippians 2:12-13
In light of what Jesus had done in humility, and in light of His exaltation, the Philippian believers were to work out their own salvation with fear and trembling. In other words, they were to live their lives in such a way that was becoming a people saved by Jesus and ruled by Him. So, who does the work? Them. Us.
Paul continues by saying that the reason why they should work is that it is God who works in them, both to will and to work for His good pleasure. So, who does the work in them? The Lord.
Thus, what we see from Scripture is that the responsibility to obey Christ falls on us. But we don’t do it by our own strength. God is the one who activates our desire of obedience and Christ’s glory, and He is the one who enables our work. And in fact, that should be a motivator for us to work out our own salvation.