The spellings Ashantee/Ashanti/Asante are used interchangeably, depending simply on the fashion of the time.
A brief timeline:
1806-07 The British are drawn into the Ashanti-Fante War by refusing to hand over rebels pursued by the Ashanti
1811 The British are again drawn into a local conflict, Ga-Fante War, where the Akwapim captured a British fort.
1814-16 Same situation again. The British are drawn into the Ashanti/Akwapim War, during which the Ashanti invaded the Gold Coast area and expanded their empire to the sea.
1823-31 First Anglo-Ashanti War - In 1823 Sir Charles MacCarthy leads an invasion force from the Cape Coast after rejecting Ashanti claims to coastal areas. The "British" force was mostly Royal African Colonial Corps and Cape Coast Militia, with a handful of British officers. Maccarthy's force was overrun and routed by the Ashanti at the battle of Nsamankow. MacCarthy himself and Ensign Wetherall were killed, decapitated and their heads kept as trophies of war. The war flared on and off till 1831 when the Pra River was accepted as the border.
1863-64 Second Anglo-Ashanti War - Ended in a stalemate with both sides losing more men to disease and sickness than action.
1873-74 Third Anglo-Ashanti War - The British purchased the Dutch Gold Coast in 1871 including territory claimed by the Ashanti empire. When the Ashanti invaded, Britain finally lost patience and sent a sizeable force of British regulars for the first time. Sir Garnet Wolseley was appointed to command a force of approx. 4000-5000 men including a Naval Brigade from the West Africa station. The British marched from the coast into Ashanti territory, and fought many engagements including the battles of Amoaful in Jan 1874 and Ordahsu soon after. The Ashanti capital Coomassie (or Kumasi) was briefly occupied and the Asantehene (Ashanti king) forced to sign the harsh Treaty of Fomena in Juy 1874 to end the war.
1895-96 Fourth Anglo-Ashanti War - Wanting to keep French and German influence out of Ashanti territory (and off its gold), the British were anxious to conquer Ashanti once and for all. A British expedition left Cape coast in much the same way and following much the same route as in 1873-74 and arrived in Kumasi in Jan 1896, after encountering very little resistance. The Ashanti were directed by the Asantehene not to resist. The king was arrested and deposed and the Ashanti leaders exiled to the Seychelles.
1900 Fifth Anglo-Ashanti War (or War of the Golden Stool) - The remaining Ashanti court not exiled to the Seychelles attacked the British Resident and other representatives at the Kumasi fort. The attack was soon defeated. The once proud Ashanti Empire was absorbed into the Gold Coast colony on 1 Jan 1902, and is nowadays an integral part of Ghana.
That's all for now!