I would not use a 10 gallon for a sump in a 120 gallon tank. I would put the largest sump you can fit under the stand/afford. I would say a minimum 20 gallon long for a sump, but something more like a 40 breeder might be better suited for that tank. The larger the sump you have the more total water volume you will technically have, and this leads to a more stable tank. I have a 10 gallon sump for my 55 because its the only tank I could fit under the stand, and I think it's too small. The larger sump will also give you more room for filter media to harvest beneficial bacteria for your tank.
there are plenty of DIY sump threads and websites online just do a google search. They all serve pretty mucht he same purpose it all just depends on how simple or complex you want to go. Its typically a good rule of thumb to have 8-10 times the turnover rate of the aquarium, so your 600 gph pump would be a bit undersized (I use a 600 gph pump in my 55 gallon). You would want to have a pump that does closer to 960 to 1200 gph to achieve this. However, you will need to account for head loss from the water column travelling from the sump up to the tank so the pump should ideally be slightly larger than this.
Is your 120 drilled? If it is not drilled you will need to build or buy an overflow or drill the tank. If you do buy an overflow it will also need to be large enough to facilitate your flow rate (which will be given with any overflow you look at purchasing).