How do I add the integers −8 + 5 − 3 in the correct order?

Work left to right, one step at a time, and keep track of the sign. Start with −8 + 5. Here you are combining a negative 8 and a positive 5; since they have different signs, subtract the smaller size from the larger: 8 − 5 = 3, and take the sign of the bigger number, which is negative, so −8 + 5 = −3. Now do −3 − 3. Both are negative, so add their sizes: 3 + 3 = 6, keeping the negative sign, giving −6. So −8 + 5 − 3 = −6. A useful picture is a number line: start at −8, move 5 steps right to −3, then 3 steps left to −6. Moving right adds, moving left subtracts.