Identifiying Parallel And Series Connections

That depends on the points you are interested in. There is no parallel or series connection for all setups in this sketch.

An analysis of the resistance between a and b will look different from an analysis of the resistance between a and c, for example.

 

1) Don't be fooled by the drawn orientation of components; diagonal, horizontal, vertical --- makes no difference at all. The only thing that matters is what they connect to. That is to say, Topology counts, not artwork.
2) The same goes for wires: A wire that wiggles around the page and jumps over other things is theoretically the same as a point! What matters is what it connects together, and all those connections can be thought of as being to a single point.
3) You are free to rearrange the diagram in any way you want so long as you don't disturb the topology (what is connected to what). You can stretch or shrink wires, reorient components, slide connections along wires, etc., to your heart's content if it will make the circuit more obvious to you.

In your first circuit, which I've reproduced here:



Note the points I've marked C and D in red. There's no reason why that wire has to be that long. And there's no reason that those resistors connected to it have to have the connection points where they're shown. The same goes for the wire at the bottom and its connections.

Why don't you redraw the circuit, bringing the connections of those resistors to the center of those those wires? (Slide the connections of the 4 and 12 Ohm resistors to the middle of the wire CD. Do the same for the pair of resistors at the bottom with its wire. Can you spot any parallel or serial opportunities now?

For the second circuit, try to imagine what it would look like if your were to "pick up" node d and pull it over to lie between nodes a and b. Don't be afraid to bend and twist the wires or reorient the components. Just don't break any connections!

Note that not every circuit will present you with opportunities of parallel or series components to simplify, no matter how you rearrange the diagram. In those cases there are other methods you will have to apply, which I'm sure you'll be learning about soon.