My ideas on what fittings you have in your current setup and what to remove for installation of your new faucet:
BIG CAVEAT - I am just a DIYer, not a plumber. I could be wrong in part or all of this analysis.
What you have
The connection in back seems to be 1/2" threaded water supply with gray (plastic?) tee fitting allowing a plastic hose to split off to side (perhaps for dishwasher?). The top connection (in green) on the gray plastic seems to be a standard 1/2" (?) threaded male end with a standard flexible faucet water supply hose connector threaded onto it.
The connection in front is harder to distinguish - more photos with better focus/details of the connection would help here. There seems to be a standard rigid copper faucet water supply line running upward, connected (in red area) with what looks like it could be a standard 3/8" compression connection. This would be a real metal compression connection, with rigid metal nut covering a metal ferrule compressed between the nut and threaded section.
The blue area right below that looks like it is part of the compression connection, but also threads onto the 1/2" (?) water supply pipe coming up from below. I believe the blue part is an adapter with 1/2" FIP threading on the bottom and a 3/8" compression connection on top.
What to remove - back connection:
First, shut off water to this connection (may require shutting off house). Open faucet tap to release residual water pressure at connection. Be ready for water to drain out of loosened connection.
This seems to be a standard supply hose connection. To remove, hold gray block below gently but securely in place (with appropriately-sized open ended or crescent wrench) and use another appropriately-sized wrench on the hex flats of the flexible hose connector to unscrew it. You do NOT want the gray fitting to turn relative to the incoming water pipe below it.
You should be left with a standard 1/2" MIP threaded connection to hook up as described in my main answer.
Possible gray block removal - back connection
Update: Apparently the gray block tee goes to a separate filtered water faucet, no longer desired.
Simply unscrewing the gray block from the lower water supply pipe should indeed leave a 1/2" threaded connection, suitable for new faucet connection as described in my other answer. It's a bit tricky, as you don't want to damage the lower incoming pipe. Use a pipe wrench (3 points of contact) or small strap wrench to hold the lower pipe from turning while you use an open-end or crescent wrench on the flat sides of the gray block to unscrew it.
What to remove - front connection:
First, shut off water to this connection (may require shutting off house). Open faucet tap to release residual water pressure at connection. Be ready for water to drain out of loosened connection.
The goal here is to unscrew/detach the part in red -- the upper rigid copper faucet supply pipe and the nut (and hidden ferrule inside) WITHOUT disturbing the part in blue -- the adapter.
To remove, hold blue adapter gently but securely in place (with appropriately-sized open ended or crescent wrench on its hex side flats) and use another appropriately-sized wrench on the hex flats of the smaller red compression nut above to unscrew it. You do NOT want the blue adapter to turn relative to the incoming water pipe below it.
You should be left with a standard 3/8" threaded male compression connector. The 3/8" compression female connector from your faucet's flexible supply hose should directly screw onto this. Remember, finger tight, then perhaps 1/4 turn further with a wrench, since there is a rubber washer inside to make the seal.
