I have a project in which I am making an 8-bit computer from TTL chips, I am using some SN74LS08N AND gates in the ALU. I know to set an input to HIGH, you must leave it unconnected, but for LOW, it must be connected to ground. How can I control the input of an AND gate with the output of an SN74LS32N OR gate? I don't think I can use an inverter for this, because the inputs of that would require the same problem, I think.
For the same logic family, 74LS in this case, tie an output directly to an input. It doesn't matter what the logic function is, output ties directly to input. For the 74LS family, one output can drive 20 inputs.
You are thinking of unused inputs, and you're in for trouble. Yes, you tie an input low by connecting it to ground. For a high, though, you must not let it float. The easiest way is to use an unused gate, such as a 74LS00 or 74LS04, whose input was tied low. Another way is to use a pullup resistor. 10K to +5 will work just fine. If you let the input float, a 74LS gate will usually work, but every once in a while it will act like an antenna, and change state at apparently random intervals.
While 74LS will usually be OK when floating, the same is not true of CMOS logic. ALL unused CMOS inputs MUST be tied either to ground or power. And I'm not speaking hypothetically here - trust me.
The alternative to using another gate in another IC is to use a pull-up resistor connected to +5V. There is a previous question related to this...