This article relys heavily on the four part article:
"What Kind of FM Antenna is Best for You?", by Michael J. Salvatti, appearing in Audio Magazine January through April 1978
if you haven't done so already - you might want to read the first part of this article (FM Antennas). It contains several definitions that you will need.
Your task once the antenna is installed is one of getting the signal to your tuner with the least amount of losses along the way. Every dB of gain was won at the cost of at least an element or two, and boom length. You should think long and hard before giving up the hard won gain needlessly. True - the height of the antenna and the noise rejection bue to the beamwidth are also important factors, but your tuner will generate a fixed amount of noise itself. Your tuner's sensitivity is largely a function of the amount of noise in its front end. If your received signal dips below this noise level, all you will hear is static. Gain helps you fight that inherent noise level.
---- And some things just just plain don't work -----
Signal splitters are becoming a thing of the past (thankfully). The varactor tuners of most cable ready televisions do all of the band splitting that is required. That is because television is no longer just "VHF" and "UHF". It now has almost continuous coverage across the RF spectrum to accomodate the various cable bands (that are transparent to the user). The only connector you will find on most televisions now days is the 75 ohm cable TV type. In years past, however, there would be two screws for a VHF antenna, and two for a UHF antenna (or some commbination of screw terminals and cable connectors). TV antennas might be separate - one for VHF, and one for UHF - or combined, with a cute little UHF antenna tacked onto the front of its big brother VHF antenna. That type is still commonly sold in stores. If two separate antennas were used, two lead-in wires were also used, and routed to the right set of screws. But - if one antenna is used, there is only one lead-in wire, and the antenna output must be "split" before being connected to older TV's with two sets of terminals.
So - where does FM come into the picture? From the "rip-offs that doesn't work" department: there are splitters that have a third set of terminals - for FM. Assuming the TV antenna will actually work on the FM band (see my antenna article) you might think this is a good thing - just run VHF to VHF, UHF to UHF, and FM to FM and everything will work. Right?! WRONG! FM is too close in frequency to the VHF low channels for a cheap filter to be used. The splitter will cause some very serious losses on low VHF (channels 2-6), FM, or both. You will be much better off buying a two-way splitter, and running VHF to both your TV VHF and your FM radio (and UHF to UHF). FM, after all, is sandwiched in the middle of the VHF band.
There is another type of splitter that is used to route signals from one source (usually the TV cable) to several destinations (or TV sets). These can be used to split the signal from an FM antenna to more than one tuner, but at the loss of 3dB or more. It might be better to forego advanced DX'ing on one of the tuners in favor of the best (or most convenient) tuner.
Antennas, transmission lines, and tuner inputs all have characteristic impedances. You absolutely MUST match these, or losses will occur. Fortunately, there are only two impedances commonly used in home equipment: 300 ohm and 75 ohm. Antennas are almost always 300 ohm, and tuners are becoming almost universally 75 ohm. Older tuners had two screws for 300 ohm, however. The flat wire with two conductors separated by about 3/8 inch is 300 ohm. Cable TV wire is 75 ohm.
If you have a 300 ohm antenna, 300 ohm lead-in wire, and a 300 ohm input on your tuner, you can just hook it all up directly, and the only losses you will have are the losses in the wire. The trade-off here is that the wire will tend to pick up interference (and stations out of the beamwidth of the antenna). If noise and / or interference is a problem, you are going to have to use sheilded lead-in. 300 ohm sheilded lead in was available in past years, but I don't know if it can be found now. This means you are going to have to consider the use of 75 ohm cable, and inserting two matching transformers into the "chain" - one at the antenna and one at the receiver.
If you have a 300 ohm antenna, and 75 ohm tuner, you must use at least one matching transformer. You might as well take advantage of 75 ohm cable in the process. It has lower losses than 300 ohm twinlead, and has better performance when it gets wet. There is little to be gained by using 300 ohm twinlead from the antenna to the tuner, and converting it at the back of the tuner.