@cptartapp do you know why it increases on the sale only? I believe it was because in rates days if you improved your property the charge went up immediately which meant people decided not to extend or improve their houses or those of their tenants. I used to work in council tax.
@Rojo019 just research before you do it. In a street of identical houses it is fairly easy to band (when they rushed in council tax following the poll tax) whereas if you have an eclectic mix of properties or a row that all seems to be 3 bedroomed houses but instead has some 2 bedroomed houses in there then banding can be wrongly applied.
Best starting point is the VOA (Valuation Office Agency) who actually band the properties, the council produce the bill. Have a look at your street on the VOA's website here https://www.gov.uk/council-tax-bands there will be markers to indictate if a change will be looked into. Look at neighbouring streets too. Try to find houses the same as yours, google maps makes this relatively easy these days with both satellite and street view so you can see plots are roughly the same size too.
On a new build estate we used to band a 3 bed detached as a D band, 3 bed semi as a C, 2 bed semi or row house as a B, 4 bed an E, "executive" 4 bed usually double garaged and a 5 bed as an F. Everything is worked off a D band, ie a C is a percentage drop on the D band etc. This does not apply to bungalows, they are usually a one band increase on what you think. It has been 20 years since I did that job but this stood where I worked as a rough guide.
Council tax is a hereditament so is a combination of the land it sits on 50% and 2 adult occupiers the other 50% hence the 25% discount for sole adult. So if a house comes with a lot of land it will probably have a higher band than you think because it isn't just the house on it.
And good luck, I hope you do get a band reduction, because it can go either way, ie your's is correct and everyone else gets increased so if I were you I wouldn't tell anyone you were doing it 