Well ... since others already did, I'll also note the obvious solution is to cut down the household background noise. But, of course, that's (also obviously) not much help if you can't.
What a noise gate (either a hardware one or a software plug-in) does is really just to mute the channel when the signal is below some level. When you're actually singing, it doesn't do anything to the noise: it's still there, just "behind" the sound of the singing, so not very obvious. Depending how you set up the gate, it will mute the noise between phrases, or between words or syllables. You can do that with a software plug-in after the track is already down, and it will be (a) more adjustable, (b) more reversible and (c) probably more accurate, in terms of not cutting off the tails of words and the like. Of course, if you want to be laborious about it, you can also do it by editing the track by hand (don't do that ... it's fine for muting a track between verses, but between syllables?).
Whether there's a noise-reduction plug-in that can reduce the noise that's under the vocal (rather than just between syllables), I don't know. It's technically doable in fairly obvious ways when the noise is something regular and periodic, like a hum. If it's dogs walking around, probably not so much, except by doing things that may have a noticeable effect on the signal you're trying to preserve.
If it were me, I'd probably:
- Use a relatively directional cardiod mic (probably obvious).
- Work the mic close.
- Maybe try double-tracking the vocal. I like to do that with my own voice anyway, but whether it's a good idea depends on a lot of other things. For good/highly expressive singers, it may not be a great idea. For singers who are shaky in terms of pitch and tone, it might be beneficial for non-noise-related reasons).
- Run a plug-in that works like a noise gate on the signal after it's recorded, and spend as much time as is warranted to feel around to get it to work well.
- Possibly run a "noise-reduction" or "noise-cancelling" plug-in and see if it helps. It won't get dogs (or probably the "whoosh" of air ducts, other than my cutting all high frequencies), but if there were the hum of a furnace motor, it might be helpful.
- Not get too obsessed about making a pristine recording. After all, it's not a string quartet. Uhm ... it's not a string quartet, is it?