By the direct reading of the move it would seem that it's limited only to commune and cast a spell moves, but I am not completely sure.
The move is very explicit: the Ranger gets only those two Cleric moves.
You can't advance as a Cleric, but...
A Ranger who takes God Amidst the Wastes is still limited to selecting from Ranger moves as they level up. The text
Every time you gain a level thereafter, increase your effective cleric level by 1.
refers only to boosting the level-based cap on spells you can prepare when you Commune.
The Ranger also has pretty much no access to moves which could let them multiclass, and therefore take advantage of the Cleric spells. Advanced Cleric Moves like Invigorate, Serenity, or Empower all depend on Cleric spells being there to begin with.
There are, however, two options.
Option 1: Beg
I mean, that's just the rules, right? If you're really playing this "servant of a wilderness god" thing to the hilt, your GM might be fine with you taking Cleric moves anyway as you level up, instead of Ranger ones. Breaking the rules to suit the campaign you've got going on is just part of the job.
But it is breaking the rules, so y'know. Don't depend on it, use at your own risk.
Option 2: Spirit Animal
The Ranger does have a 6-10 move called Special Trick.
Choose a move from another class. So long as you are working with your animal companion you have access to that move.
And pointing that at a Cleric advanced move might seem a little weird? But if you are playing into the "servant of a wilderness god" thing, it wouldn't be that odd for your animal companion to have some effects on your cleric spells.
Level Up is a Move. In the Dungeon World context, Moves are just a rule with a trigger.
When you have downtime (hours or days) and XP equal to (or greater than) your current level+7, you can reflect on your experiences and hone your skills.
- Subtract your current level+7 from your XP.
- Increase your level by 1.
- Choose a new advanced move from your class.
- If you are the wizard, you also get to add a new spell to your spellbook.
- Choose one of your stats and increase it by 1 (this may change your modifier). > Changing your Constitution increases your maximum and current HP. Ability scores can’t go higher than 18.
As you can see, you can only select new moves from your class. Since God Amidst the Wastes doesn't change your class, you can't take advanced moves from that class when you level up. Bard, Cleric, and Fighter provide the Multiclass Initiate and Multiclass Dabbler moves, which would allow you take a prerequisite ability and then a subsequent ability but even they don't fully unlock the playbook. The only way to do this is to take the Beyond 10th Level option Change entirely to a new class.