The Hyderabad - Hampi route is nice, with couple of bad patches here and there. I have done this route recently. Since you're planning to visit Hampi, lemme tell you you will have to cross the worst possible strech of around 1-2 Kms. It a NO-ROAD section which you would encounter around 20 kms before Hampi. Doing this section at night is not at all advisable.
Anyway the route to Goa directly from Hyderabad is advisable as below.
Hyderabad - Jadcherla - The 4 lane Bangalore highway - No comments needed for this strech! Excellent !! - 88Kms
Jadcherla - Mahbubnagar - Raichur - SH 13 Not so bad road - Some craters available and are visible at reasonable speeds - 80 Kmph. 1 + 1 hump per town you pass through till Raichur. - 135Kms
Enter Karnataka, the road its bliss. They way Karnataka state government maintains roads is commendable. The SH 20/14 from Raichur to Belgaum is excellent. Two lanes but maintained well can do good speeds. Negative point - expect 3 + 3 rumblers per town you pass on this stretch. - 356Kms
Belgaum - Amboli (includes Ghat Section) - Sawantwadi - Panjim makes better sense than attempting Londa - Ponda stretch since the latter is abused by dump trucks. Two lanes again but excellent. - 180Kms
There is one more route via Solapur, no idea on it.
For Hampi, Follow this route:
Hyderabad - Jadcherla - Mahbubnagar- Raichur- Gangawati- Hampi.
i used to regularly drive from belgaum to hyderabad. another route concerning this stretch is hyderabad-sangareddy-homnabad-gulbarga-bijapur-belgaum-anmod-goa. road is very good,less trafficked . road between belgaum to goa via anmod is not safe at night as the mining tippers are very active and rash during night time. road belgaum-sawantwadi-goa is better and safe for night drive.