Some basic (and generally essential) steps in the costuming process:
Read the script. Establish who the characters are, what the time period is, what the locations are, the times of the day, the season (summer, winter), etc. Identify any specific requirements the script has for costumes.
Research. Get a feel for what has traditionally been done for this and similar shows. See if there is a specific look that audiences will be expecting.
Meet with the director and the rest of the production team. Find out what the director's vision for the show is, and what the set designer and tech designer might be considering that would impact the costumes (i.e. Stairs can be difficult to navigate for characters in large dresses and high heels. Share your preliminary vision for the costumes.
Refine the ideas in your head based on what you learned from the meeting.
Get to know the cast you will be costuming. If you will be getting costumes custom-made for specific actors, you will need to take their measurements (don't rely on what they tell you!). If you will be renting or buying costumes, measurements might be less important. If you know the rough height, weight, and build of your actors, you can usually make a pretty good guess what will fit them and what won't. If you need to find shoes for your actors, try tracing the outline of their foot and/or their everyday shoe on a piece of paper. You can take that piece of paper with you when you're looking for shoes, and place possible shoes on those outlines to see how well they match up.
Find out what your budget is, and what your options are for working within that budget. Where can you get quality costumes for cheap? Where do you have room for flexibility? Can you change the time period of the show to fit what is available for rent? Helpful hint: Allow for a $1,000 budget for every 10-15 actors in your cast. That's about you can expect to spend. Less than that is ideal, but generally not very realistic. If you only need one costume per actor, and if the costumes are simple dress pants and button down shirts and whatnot, you may be able to do it for a lot less. If your show is more complex, more show specific, like Little Mermaid or Shrek, you can may end up spending a lot more. Anything in between will probably land around that general rule of $1,000 per every 10-15 actors. Remember, each costume includes multiple pieces, like dress, tights, shoes, hat, coat, gloves, etc. Each actor usually has more than one costume. Each costume item typically costs at least $10 to rent, buy, or make. Dresses typically cost more ($20-30 minimum, and lot more for the fanciest, most iconic ones). Don't forget to include room in your budget for repairs, replacement of damaged or lost items, and dry-cleaning expenses for after the show.
Start pulling costumes. Renting and buying are typically your best options, because they are generally quicker, cheaper, and if you rent you don't have to worry about storing the costumes afterwards. Give yourself plenty of options. If you're renting, pull multiple options for each character if possible. Items like dress pants and button down shirts that are time period neutral and can be used for dozens of shows are great to buy from places like Goodwill. These places typically don't allow returns, but you can buy pants and shirts for around $5 each, and if something doesn't fit you can probably find a use for it in a future show.
Start trying costumes on the actors. Don't wait until you have everything. Bring things in to rehearsals as you get them, so that you have time to make adjustments to the fit/size, or return the item and look for something else. Take pictures of the actors in the costumes when they try them on, so you can show the director and other production staff, to ensure that your vision is lining up with theirs. Pictures are also helpful when trying to keep track of which items worked for each actor.
Once you have tried costumes on and found at least one full outfit for each actor, do a costume parade. You'll want to see the leading and supporting characters individually and in small groups (couples should be seen as such). Ideally, you'll also want to see every collection of people that share the stage at any point in the show. For example, if man 1 is on stage with townspeople and several other characters, you'll want to see that whole group together, to ensure that their costumes all fit together nicely. Chances are, when you see people in groups, you'll find that some of the colors on the costumes don't work, or some of the patterns may clash or just be too "busy", or the time period of someone's costume may stand out as being distractingly different than the others, or a poorer character's costume may outclass that of a rich one. Make sure the director and some other people are present at the costume parade, so that you have several pairs of eyes to spot potential problems. Make sure to write down notes about undergarments and accessories. For example, if an actor will need black socks, write that down. If an actor's costume is too light, too see-through, they may need a white or flesh colored camisole or undershirt. If they are wear a strapless dress, they'll need specific undergarments to go with it. If somebody's zipper is problematic or the costume is otherwise difficult to get on or off, take a note of that, because you may need to find a different costume option (especially if that actor has to change into or out of that costume quickly). Also consider personal hygiene. You may need to make notes or plans, due to things like body hair, excessive sweating, the heat of the stage lights, makeup stains, etc. Light colored and solid colored costumes can be particularly problematic when it comes to sweating and makeup stains. This might also be a note for the lighting designer. If you need to use costumes that are see-through or will be covered in sweat, makeup, etc, you probably don't want to have the brightest, sharpest lighting settings for that scene. Same thing goes for sound. If certain costumes are too noisy, that could be something you need to work out with the sound designer. If actors have to wear hats or glasses or fake beards or other head/face items, that might effect microphone placement, makeup design, hair design, etc.