A deck list (or at least the spells) would have been useful, but here's how I normally design a basic/shock/fetch land mana base:
1. Get enough basics to cast all your spells
I assume that black is only in your deck for Rhinos, and that you have heavy white and green components.
2. Get enough shock lands to cast all your spells
Normally about 4-6 shock lands, unless you have a reason to add more fetch lands (such as delve cards). Again, since Black is your least important color I would prioritize Green and White.
3. Get enough fetch lands to cover your basics and shocks
You have 10 total, so get 10 fetches. Each fetch can find each shock that you play, so you choose fetches based on the basic lands you'd like to draw first. Green is your most important color, so you prioritize that.
4. Add utility lands
Based on your list.
5. Check number of lands and number of sources
We have 24 lands now, while we want 23-24 (since we curve out a 4 mana for Liege and Rhino, and want to do so fast). We have 17 Green sources, 17 White, and 16 Black. Especially if you're playing something with GG (such as Eternal Witness) I would prefer a bit more green mana. However, since we only need Black for Rhino we can go down to ~12 Black sources. (see Frank Karsten's article on mana bases). White seems fine.
Based on this I would remove a few fetch lands with black and add a forest and some GW lands (Thickets).
6. Start playing with this, but keep an eye on it and be ready to adjust.