Is there a better way to get to Southampton from the Waterloo area other, than the obvious, railroad?
We would be travelling on a Sunday, if that matters. Thank you.

Nope - train is fastest, best and usually cheapest

"Better" is a rather subjective word, but in this case the answer is nearly always going to be No.

It might be slightly quicker to take a taxi or prebooked car service, but the time difference would be negligible, and the cost is much higher.
It would probably cost less to take a National Express coach (and you book far enough in advance) , but you'd need to get to Victoria bus station, and from there it is a 2.5h hour trip. If saving £20 is worth the extra hour/90 minutes journey - and you don't have a lot of Luggage, this might work for you.
I'd still use the railways - There are two direct trains an hour from Waterloo (plus one that needs a single change at Basingstoke) that cost around £40 and take 90-100 minutes.
The only proviso to this is that you need to look in advance as to what rail maintenance is planned - Sunday's are often used for routine repairs, and there may be the dreaded "Rail replacement bus service".