If you want a decent journey, look at going via London. However cheap AP fares will have sold out for this journey weeks ago, so I doubt that would be a viable option now.
So you are probably stuck with the abysmal service by Cross Country, although if you time it right you can get a real train and therefore probably a seat, from Derby to Bristol.
The full fare (no railcards) is £60.50, so just try splitting at various places. e.g. NOT-CNM is £37.60 and CNM-BTH is £19.00, therefore that's down to £56.60 already. Cross Country's fares are generally the worst value in the country, so if you can find fares set by other TOCs then that lowers the cost.
The cheapest walk-on method of going via London is Nottingham-Stroud (£42.00) plus Swindon to Bath (£11.00) totalling £53.00 however you will struggle to get through the LU barriers as the staff are poorly trained and don't understand what the Routeing Guide is, they are 'programmed' only to look for a "+" symbol which isn't on this ticket. Bring the relevant documentation from the RG if using this ticket. I did it once and was only questioned at Kings Cross St Pancras on LU (Paddington is not a problem as the LU barriers could be avoided, and I think they have been removed completely now and replaced by FGW ones), after a 'discussion' I was allowed through.