Home studios by their nature tend to have limited resources. It's not unusual to have one all-purpose microphone. Vocals that really sparkle tend to be recorded with large capsule condenser mics, and this type should be your first choice. Each brand and model has a particular sound. No one mic is the best for every voice, so if you have the option, try several, regardless of type. Let your ears guide you.
It is always good practice to record vocals with little to no equalization, compression or other signal processing. Adjust the mic and performer. Save other sonic changes for after recording. An experienced vocalist will use technique to control her singing level; a novice singer may need the engineer to lower and raise the volume of the mic to keep the recording from being too loud or soft. The only exception is to cut low frequencies below 80 hertz. This will reduce low frequency rumble that may be picked up by a mic stand. Many mixers have Lo Cut filters for this purpose.
Home studios are often placed in rooms with parallel surfaces that present compromised acoustics. Too much room-sound reflecting into the microphone at the time of the recording is difficult to mask. Improvising an isolation booth is a good idea. A blanket hung over an extra mic stand set to make a "T" shape and then placed behind the microphone will absorb sound energy. The vocal sound will be more intimate and dry. Reverb can be added later, if required.
First, listen for frequencies that are objectionable, perhaps nasal tones or excessive "ess" sounds. Experiment with EQ settings, removing frequencies first, rather than adding. Boosting a band of an equalizer increases noise as well, so cuts are preferred. Second, evaluate how the vocal sits with the rest of the instruments. If, for instance, a guitar is clashing with the vocal, identifying and cutting clashing frequencies in just the guitar track may stop the struggle.
Most vocals have some degree of compression to condense their dynamic range. This in turn can add punch, causing the vocal to stand out. Moderate amounts of compression are generally preferred unless a special effect is desired. Keep ratios of compression low, attack times moderate and release times long to minimize compression artifacts. The compressor may have an effect on EQ. Re-evaluate EQ and adjust further, if necessary.
Use of reverb is subjective. Tastes change, and yours will be your guide. Try to listen to your vocal objectively to decide if there is a sonic reason to add reverb. Perhaps your vocal sounds great but it is out front of the instruments too much, and lowering the level buries it. Try a bit of reverb to "move" the singer back into the band. Reverb gives a sense of distance to the human ear.
A time-honored way of thickening a vocal and covering flaws is to have the singer sing the same vocal twice. Tracks are played back simultaneously. Artificial double tracking (ADT) is a time and pitch effect common in many signal processing devices. This can be used to create a similar effect without the singer matching her performance.
Typically lead vocals are panned to the middle, but there is no law. Try panning the vocal to the center, but panning the reverb on it to one side or the other. Double tracked vocals can be panned to each side of center. Likewise, panning of instruments may be just the step needed to make room for the vocals.