Yes, aside from checking with a tester to make sure the circuit is actually off... there are several subtle issues that frequently catch novices.
Don't remove screws all the way - they are captive. After some distance they will start getting stiff. Stop there, don't force them.
Unless you're using the new "lever-nut" style receptacles or the dreaded backstabs, you need to torque the receptacle screws if they specify a torque. This requires a torque screwdriver; a recent requirement in NEC 2014 due to recent science that says it matters. (and some other field testing at trade shows that revealed pro electricians can't torque "by feel" any more accurately than their spouse, apprentice or other traveling companion.)
Receptacles - Tabs
Most common North American style receptacles have 2 sockets and 2 attachment screws on each hot and neutral side. Between the screws is a "tab" of metal that can be flexed back and forth and snapped off.
Check the tabs. If your old receptacle has any tab(s) broken off, you must break them off on the new receptacle.
There's your tab. This $1 outlet was ruined, by soldering (!) and removing the screws.
If you break one off by accident, and the receptacle is valuable, that's fixable.
On receptacles, brass screws are for hot and silver wires are for neutral.
Broken tabs and GFCIs, USB or other specialty outlets
If your have an outlet with tab broken, and you are trying to replace it with a GFCI, USB or other specialty outlet, this is a specialty task. You'll need more research. Don't just cram the wires on anyway!
GFCI receptacles otherwise
Take care to distinguish LINE from LOAD.
If you are replacing a plain outlet with GFCI, leave the warning tape on, and hook up only the LINE terminals (cap off spare wires). Power up the circuit and test the GFCI. After it works, shut off the breaker and consider adding wires to LOAD if you want to use this GFCI to protect that downstream part of the circuit, which is a cost-saver if you know what you're doing. Otherwise, add those wires to LINE. Most GFCIs support screw-to-clamp for 2 wires per screw; just read the instructions.
Switches - 3-way (UK: 2-way)
Anytime you see 3 or more screws (besides ground), watch out. 3-way switches are tricky, because screw position and wire colors are meaningless - they vary on every switch. The color of the screws matters. (or if uncolored: labeling).
- One screw will be black, and is called "common".
- The two other screws are brass colored, and are "travelers".
I'm a big fan of buying a 5-pack of colored tape, and marking both travelers with yellow tape. (for a second 3-way: blue tape). This makes it a lot easier to tell what's going on in a busy box. There is no need to distinguish them from each other, so use the same color. Two same-colored wires in the same cable tells an experienced person these are travelers.
Marking is helpful because the native wire colors of travelers mean nothing. I've seen 3-way circuits in a how-to guide where 3 different pairs of colors were used in the same 3-way loop!
4-way (UK: 3-way)
These switches have 2 brass and 2 black screws. Two pairs of travelers attach. One pair of travelers attach to the brass screws. The other pair of travelers attach to black. In most cases, it's easy to group them in pairs - any pair of travelers is always in the same cable or conduit. Just make sure that is so.
For marking, I typically mark all 4 travelers yellow, since you can easily see the grouping by looking at the cables.