Hi, I live in Puerto Rico. El Yunque is the best area to go hiking on the Island. There are very nice trails (most connect so one can make some nice loops when connecting them with the road) with solid elevation gain. Be prepared for rain at any time. Water is readily available everywhere (most trails are alongside and/or cross streams at several points) but you must filter it. For the foreign traveler the heat and humidity is unrelenting at lower elevations and never really eases up until the very top when you are out of the cloud forest for a moment and exposed to the wind. Expect 70-100% humidity and temperatures in the 70-90F degree range and most times zero wind due to the thick vegetation except in clearings and at the top. (synthetic clothing only, my choice is MH Canyon pants and light synth shirt plus a breathable stormproof shell with pit zips and vents, smartwool socks and 5.10 camp fours or Vasque goretex-boots)
Going off trail is not recommended, ground conditions are slick and full of organic debris and composed mainly of exposed roots and eroded soil. It is REALLY slow going at best. You cant see the terrain (cliffs, drops, ravines) until you are right on them due to the vegetation. The only time I have gone off trail is while trying to go up Juan Diego Falls (and stream). It was fun but slow going due to slick terrain but fun.
Here is a map of the upper trail system (http://www.elyunque.com/maptrailsbig.htm
The trails are mostly compacted gravel with rocks imbedded in the soil (most are slick, covered with moss) with some old rough concrete mixed in at times for stability (specially on the lower most popular trails), there are some old concrete and rock steps in the mix. There are some parts that might have a little mud due to erosion but overall its mostly hard surfaces. Beware, its SLICK coming down if its raining. Water doesnt really puddle much on the trails but sometimes flows down them so waterproof shoes are nice to have. Having shoes with sticky rubber is a plus not only for safety (more fun also!) but because it will allow you to explore some of the stream/creek beds which are bare rock and really cool! if you have time.
DO NOT miss LA ROCA, most people go the highest point and never explore this rock which is just 10min away and IMHO has the best view by far (there is quite a drop so be careful). The last 1200 ft of mountain is almost always in the clouds but sometimes you get a clearing for a few minutes, enough to see the amazing views (you are only 7-10 miles from the ocean (3400 ft below). Its best to go to the top early (before 10am) if you want a decent chance of a view but sometimes after it rains in the afternoon you get a clearing (4-5pm). Beware that if you go on a saturday or sunday the la mina falls pool and trail will be packed. The best chance for some solitude is the El Yunque Trail (my top choice, specially the top part) and La Coca Trail (lower down the mountain slightly up the road from la coca falls and is pretty much deserted all the time, its a dead end so its in and out the same way)