I'd recommend The Desert War by Alan Moorehead.

The conflict was primarily focused in North Africa however over there was also often forgotten fighting in east africa in Somaliland, Italian Eritrea and Italian controlled parts of Ethiopia. It would be difficult to give you a sufficient run down of all the fighting throughout the war on all the different fronts, but I can give a basic run down.

Italy joined the war in 1940 soon after the German invasion of France. Soon after this there was fighting primarily between commonwealth and Italian forces along the borders of their territories, fighting along the border of British controlled Egypt and the Italian colony of Libya as well as fighting along the borders of Sudan and the Italian colonies of Eritrea, and Ethiopia while the Italians managed to invade the under garrisoned British colony of British Somaliland.

The more intense fighting began in September of 1940 when the Italians launched an invasion of British controlled Egypt which was initially successful in capturing western Egypt but was driven back in a counter attack in December that was a massive success and one of the leading factors in the poor stereotype the Italians gained during the war. Britain continued driving tis counter attack into Libya into early 1941.

With an Italian defeat in North Africa becoming increasingly probably as its 10th army surrendered in February the Germans began to reinforce them with the famous Rommel commanding the Afrika Korps with the ultimate objective of taking the Suez Canal.

Meanwhile, in March British Somaliland was recaptured by commonwealth forces while Italians were defeated in Eritrea at the battle of Keren which paved the way for the later defeat of the Italians in East Africa. Native Ethiopian, Belgium and Commonwealth forces also began to make their way into Italian colonized Ethiopia over 1941 until their last major stronghold at Gondar was destroyed in November ending most of the organized resistance although some guerrilla fighting continued into the next year.

Back in North Africa the two sides were caught up in a lengthy siege at Tobruk, with it being successfully evacuated the garrison in November, elsewhere the Axis forces, increasingly lead by the Germans, put the Commonwealth back on the retreat over the course of 1941 continuing into 42 when Americans began to reinforce Commonwealth and Free French forces in British controlled Egypt with a major defeat occurring over the course of May and June thanks to the Germans breaking the American's code and partly thanks to a poorly lead counter attack.

In late July the two sides were finally brought to a stalemate at the first battle of El Alamein with the German attack at the start of the month unable to make ground but likewise the allied counter attack failed to achieve the desired effect.

Things generally went back and forth over the summer until over the late autumn a massive commonwealth lead allied attack against the Germans at the second battle of El Alamein turned the tide in the western desert war for good.

Simultaneously in western Africa allied forces invaded French north west Africa which was occupied at this time by the Germans and Italians while under the control of their puppet government known as Vichy France. This was the first major operation lead primarily by the Americans in Africa. This was largely successful with the Tunisia Campaign beginning in later November as they moved east.
In the west while in the east Commonwealth forces rapidly advanced through western Egypt and Libya until the end of the year. Meanwhile allied forces who'd landed in Operation Torch made a dash for Tunis but failed to take it before the Germans reinforced the city and they were driven off.

Some of the first major battles between American and German forces occurred in the battles of Sidi Bou Zid and Kasserine Pass in February of 43 in which Rommel defeated them in a nasty wake up call to the realities of the war and proved their comparative inexperience. However, the actual losses of men the Americans suffered wasn't catastrophic and the follow up Ochsenkopf offensive failed to achieve its goals.

In March Commonwealth forces invading Tunisia from the south east achieved a string of victories throughout March and April advancing northwards while at sea the allies began an offensive that cut off the axis forces in north Africa. In the west the American lead forces began making good headway in late April and as the remaining axis forces were squeezed the last of them surrendered in May.