The "lute" is actually many instruments. But the school of English lutenist songwriters (which includes Dowland) played and wrote for the Renaissance lute. Even this specific variety of lute came with variations: six to 9 courses were common.
Unlike the guitar, the lute is "double-strung" on all but the highest course (course being one or two strings tuned in unisons or octaves treated as a single entity). Variations (liuto atorbiatto, theorbo etc.) could be single or double strung. And there were many kinds of lutes with additional necks to support extremely long bass strings, but they belong to another style entirely.
Lutes can be gotten for fairly 'cheap', but a good one will set you back. Search for lute on the web and you'll find many makers who have web sites. Some in South America are making lutes appropriate to North American use. (there are matters of humidity that relate to instruments falling apart if you aren't careful!) They are priced quite reasonably, they ship, and there aren't a lot of problems with getting the duties paid properly. Ask about acquiring a good lute in another question, though: this question is already packed!
Many of the original printed versions of lute music are available, and one gentleman maintains an entire web site devoted to free downloads of PDFs of lute music. They are marked whether Renaissance or Baroque, so it pays to pay attention. (See the source list).
As has been said, you can tune the G string of a guitar to F# and get the same intervals as on a lute. This would be equivalent to an E bass lute. You can capo at the third fret to simulate a g lute, which is probably the range most tenors and sopranos find comfortable with the English Lutenists' music. You'll only have one string under each finger, so it will hurt more than a lute will (for a couple of reasons) so don't do this with a metal-stringed guitar, or a nylon folk guitar. Use a classical guitar which is in good repair with the action set properly. Again: This doesn't make the guitar a lute! But it gives you a chance to see (for little or no cost) if you can hack playing and singing at the same time.
The lute has gut strings and gut frets. The tension in the strings is lower than in the classical guitar---lower really than any modern stringed instrument! The frets are tied tightly around the neck, but can slip, so if you ever meet a lute in real life, be careful how you pick it up! (They are incredibly light, also, so a thumb between the last fret and the body and fingertips on the top of the lute is enough to hold it!)
It was quite common for women to play lutes, although some of the iconography (paintings) aren't meant to be of women, but rather allegories for peace. None-the-less, whole collections of music were written specifically for women to play in their chambers. Queen Elizabeth I played lute. 'nuff said. Also in the sources will be a link or two to sites of women who play lutes and theorboes.
As for singing, in the Renaissance women and boys sang the treble and descant parts (although not on stage, where it was thought unreasonable for women to be!) And no one really cares if women sing Dowland if they sing it with meaning. Emma Kirkby is certainly a woman, and she regularly gives concerts (although she doesn't play, often she is joined by Anthony Rooley or another lutenist.)
OK: The renaissance lute. Most commonly, it had 6 to 8 courses, occasionally a 9th is asked for. The tuning (using the g lute as an example) for the primary, fingered 6 courses was (bottom up) G C f a d g'. The bottom course, if alone, could be tuned to D or F; if there were two courses, they could be C and D or D and F... changed as needed. For the 9-course lute with 3 courses beyond the basic 6, the most common tuning was C D F. Sometimes you have to figure this out by looking at the tablature and judging from the sound of chords! The bottom three courses (or two or one) were fingered, but rarely.
Tablature is as mixed a bag as the lutes themselves. English tab was based on French Renaissance tab, and learning one does well for both (Attaignant, for instance). This places the top string at the top of the six lines, with a letter to indicate each fret. It is thus a schematic system rather than merely a notation: you are told quite clearly where to put fingers (i.e., behind which frets). "a" represents an open string: "b" is the first fret. For clarity, "r" is often used for "c" to make it clearly different from "e". And generally, i and j and y are used for the same fret, which would be the eighth.
Time is notated using (fairly) standard tail-and-flag symbols above the staff, and time is figured as interval-to-next-event rather than individual duration as modern music is notated.
Italian tab is just upside down from this, with the addition of using numbers on the lines rather than letters over the lines.
Additional courses beyond the basic six are shown in the space below the staff (7th course) or beneath one or two ledger lines as needed.
As difficult as it may seem to someone trained in modern music notation, the tablature system is ideal for fretted strings. After a few minutes of trying it, I'm sure you'll wonder why it isn't used exclusively for guitars and basses and mandolines and etc. Half the answer is that it is, but usually coupled with a piano staff showing time, which is the other half of the reason: time is not explicetly notated: just time between "events" (being individual notes or chords.) You're just told to hold the last event's frets until a new fret on that string is specified _or_something comes along that clashes with what you're holding.
Anyway, with this, you have what it takes to get some songs, arrange a classical guitar so it acts like a lute, and try them out. If you find you like it, ask another question about how to acquire a good lute!