If you're just starting, don't worry about beating higher level waves. Focus on stocking up on space cubes to buy permanent upgrades. Note these upgrades can be refunded so freely choose what you may need.

Captains, whichever captain you're using, max out their weapon to level 3 to give you early advantage. (For example, I start with Azurene's plasma and plan on acquiring Corrosia's slime and Raven's missiles. So I have these three captain's weapons on level 3.)

Planet upgrades, having the gold planet unlocked helps to level up faster by staying within the radius and the gold vault pick ups.

Focus on trying to get the super version of weapons as soon as possible! So try to focus on one weapon at a time. The time cube shop helps immensely. I would re-roll the time cube shop to get the weapons upgraded faster.

Pay attention to the timers for gold, time cubes, and space cubes! Pick them up as soon as possible to reduce down time. But do be careful when the waves start getting thick.

Best weapons to survive the later tougher waves are Corrosia's slime and Ram's boxing gloves. Corrosia's slime slowing down anybody trying to tap your butt, and Ram's boxing gloves knocking any fools out of your face. Would also create a swirl with Corrosia's slime to create layers of slow when the waves get especially thick and spicy near the end. Note, you do NOT need Ram in your crew slot, just his boxing gloves are enough.

For best DPS, Tesla's chain lightning, Raven's guided missiles, and Azurene's Plasma cannons are the best for clearing in that order.

Space cube chance and space cube from elite power ups will help stock up on space cubes faster to get more upgrades unlocked!

And that's basically what I've done to defeat the champion wave! Hope this was helpful.

Personally I would keep Raven and Corrosia's weapon level 3 so I can add them as crew mates with already powerful weapons ready to go.

[Update 5/12/23 My best run]
Survived almost 45 minutes with Missiles, Lightning, Plasma, and Static Field... get crazy intense at the end.