Generally speaking, if you want to connect a signal generator to a breadboard it is best to solder a BNC socket to the breadboard and use a BNC-BNC cable, if only because it is much less likely to fall off than a BNC to alligator clip lead. At frequencies above around 1MHz you shouldn't use alligator clips, and you probably want to 50 ohm terminate the BNC-BNC cable at the breadboard: this can be a 50 ohm (or 51 ohm) resistor on the breadboard itself, or maybe the component on the breadboard you are driving has a load impedance which is significant compared with 50 ohms and you need a matching network between the two.
A good trick here is to use a BNC-BNC attenuator adapter (6dB or 20dB) right at the breadboard input: the attenuation makes the cable termination relatively insensitive to the actual input impedance of the breadboard circuit, so it will always be close to 50 ohms. Obviously you need to increase the signal level from the generator to compensate for the loss. Higher loss gives better matching.
BNC 'T-pieces' are always useful to connect multiple cables together. By themselves they don't provide any matching: for that you would want a similar looking but very much more expensive thing called a 'power divider' - you probably don't need one (yet).
One or two 50-ohm BNC terminators (plug only) or through terminators (plug and socket) are handy for providing cable termination. Given your scope has switchable 50 ohm input termination you won't need to use one in that position, but it can still be handy for use with breadboards.
In general (given your scope has 100MHz bandwidth) you can happily use a conventional x10 passive probe to monitor breadboard outputs as well as internal signals. If the breadboard circuit is supposed to drive a 50 ohm load, again solder a BNC socket and use a BNC-BNC cable to the scope, selecting the 50 ohm termination on the scope input.
Also
- Don't use a scope probe (even a x1 scope probe) to connect a generator output, as they have built-in resistance and you will wonder where the signal went
- Frequency counters may have either 50 ohm inputs, high-impedance inputs (like 'scope inputs), or switchable 50 ohm termination. Use BNC-BNC cables with 50 ohm inputs and ordinary scope probes (x1 or x10 as appropriate) with high impedance inputs. Alligator clip leads and x1 probes are generally good only up to 10MHz or so
WARNING You will find yourself accumulating a large number of cables, adapters, terminators, and probes of different types. You will wonder where they all came from. And you still won't be able to find the one you need