Some of my favourites are the Letters of Fronto, which have been published in two volumes by Loeb. Fronto was the tutor of Marcus Aurelius, and there are letters between the two about their health, studying, giving speeches, day-to-day life, and the cares of running an empire. Fronto also writes to Marcus' co-emperor Lucius Verus, the emperor Antoninus Pius, various members of the imperial family and others.

Here is one from Marcus to Fronto (Loeb v.1 p. 179-181) in which he describes his day:

Hail, most reverend master. We are well. By a satisfactory arrangement of meals I worked from three o'clock till eight... Then putting on my boots and donning my cloak...I went off to pay my respects to my lord [Antoninus Pius].

We set out for the chase and did doughty deeds. We did hear that boars had been bagged, for we were not lucky enough to see any. However, we climbed quite a steep hill; then in the afternoon we came home. I took to my books: so taking off my boots and doffing my dress I passed nearly two hours on my couch, reading Cato's speech On the Property of Pulchra, and another in which he impeached a tribune. 'Ho,' you cry to your boy, 'go as fast as you can and fetch me those speeches from the libraries of Apollo!' It is no use your sending, for those volumes, among others, have followed me here... I wrote a little wretched stuff, fit to be dedicated to the deities of water and fire: truly today I have been unlucky in my writing...

I think I must have taken a chill, whether from walking about in slippers in the early morning, or from writing badly, I know not... So I will pour the oil on my head and go off to sleep, for not a drop of it do I intend to pour into my lamp today, so tired I am with riding and sneezing. Farewell for my sake, dearest and sweetest of masters, whom I would make bold to say I long to see more than Rome itself.

The joke about 'burning the midnight oil' is a recurring one between Marcus and Fronto. Marcus was Caesar by this point and was simply fanatical about work. Fronto was constantly teasing him to sleep and rest more, and Marcus was constantly talking about working through the night by lamp light. You can follow these relationships through the letters, and it is quite fascinating.

We have hundreds of letters from Marcus Tullius Cicero (106-43 BC) to his friend, the equestrian Titus Pomponius Atticus, to his brother Quintus, and to many many others, including Caesar, Marcus Junius Brutus, Pompey, Caelius Rufus, his wife Terentia and his slave Tiro. Although some of these were meant to be "published", or at least shown off to other Romans, the vast majority were meant to be private. That makes them very cool historical sources, but also means they can be hard to understand, since they're full of inside jokes and mention things without context.

They're fascinating and cover a variety of topics: Cicero's election chances for the consulship, his struggle with the tribune Clodius and Clodius' trial for sacrilege, letters of introduction and patronage, the nitty gritty of governing the province of Cilicia (including letters from Caelius Rufus demanding Cicero get him panthers, and letters from the famous Brutus asking Cicero to help him extort money from a town), collecting art and books, writing literature, trials, political alliances, economic matters, and so much more. A huge number of them come from the civil war period, and especially the year or so after Caesar's death. Oxford Classics has a great and affordable selection of them if you want to check them out!

Seneca, Nero's tutor, wrote a number of letters of a mostly philosophical nature to his friend Lucillius. I'm not much of a philosopher, but I find them perfectly charming. He's not so bothered about abstracts, but how to react to stuff you encounter on a daily basis. It's not high politics, but it is a kind of day-to-day view of one Roman life.