I've had a couple of trips to Ypres over the last few years. We stay in the Hotel Alliance in Ypres which is approx 100 yards from the Grand Place, they are happy to accomodate bikes. There are plenty of bars and restaurants in and around the Grand Place and the In Flanders Field museum in the Cloth Hall is well worth a visit. A word of caution - the streets of Ypres are nearly all cobbled, not great for riding on.
You haven't said how you're getting to Ypres. if you're driving, then parking within the walls can be a bit of a problem and expensive. However if you drive out through the Menin Gate, cross the moat and take the first right, then after about 500 yards there is ample free parking on the road side. I left my car there on both trips with no issues.
When driving or cycling in Ypres itself, pedestrians very much expect to have the right of way so be careful.
To the N/NE of Ypres the following are worth a visit.
Essex Farm - on the west bank of the canal leading north from Ypres. Cemetery and bunkers which formed a dressing station. It was here that John McCrae wrote "In Flanders Field the poppies grow..." etc. The cemetery contains the grave of Valentine Strudwick, aged 15, one of the youngest soldiers to die in the war.
In Langemark, the Langemark German Cemetery - contains statues known as the "Mourning Soldier" and three original German bunkers. The Harry Patch Memorial is on the bank of a small river, it's a small memorial commemorating where Harry Patch fought.
In nearby Poelkapelle - there is a memorial to French pilot Georges Guynemer in the form of a column with a stork on it. Passendale - not a lot here but the Carolus Bar sells Passchendaele beer, the bottles make a good souvenir. About a mile SW of Passendale is the Tyne Cott cemetery, this is a must visit for its sheer size if nothing else.
To the E/SE.
About 4 miles due E of Ypres is Polygon Wood. in it's NW corner, there is a bar called Cafe de Dreve, which contains lots of WW1 exhibits. It's proprietor is Johann Vandewalle; he is a mine of information about the area and WW1, having appeared quite regularly on TV, and will happily talk to you.
Roughly two miles SW of Polygon Wood is Hill 62. As well as the cemetery, there is a small privately owned museum to the rear of which are some preserved trenches. The museum apparently hasn't changed much in 80 odd years but there is some debate about the whether the trenches are original or reconstructions.
About a further mile SW is Hill 60, there used to be a museum here [closed when I was last there] but there is an area of cratered land with a couple of bunkers.
Leaving Ypres to the S, via "Shrapnel Corner" [a roundabout], you can follow the main road [not too busy] through St Eloi, passed Wijschate ["Whitesheets"] to Messines, along the top of the Messines ridge. In Messines there is the Vivaldi Cafe [well worth a visit, they serve Duval] and the Ulster Tower. The road drops down from the ridge and continues south to Ploegstraat ["Plugstreet"]. On the way into Plugstreet you pass the Hyde Park Cemetery with a large memorial and two stone lions. Opposite this is the much smaller Hyde Park Cemetery Extension. Buried here in the front row [of about five] is Ronnie Poulton-Palmer, England rugby captain and heir to Huntley and Palmers biscuits. Also buried in the front row is Rifleman S McBride, who was executed for desertion.
There is a bar next to this and behind them is Plugstreet Wood. As Canardly said, this is well worth going into just to experience the atmosphere, I'm sure that the temperature drops by several degrees as you go in. If you retrack back up the main road towards Messines to St Yvon there is a track on the E side of the road, signposted to Prowse Point Cemetery, Mud Corner Cemetery etc etc. This is the best way into the forest but these are at best farm tracks, at worst muddy footpaths. Look out for eroded trench lines and derelict bunkers in the woods.
If you return to the main road and head S again it will take you through Ploegstraat all the way to Armentierres if you want to go that far. In Ploegstraat there is a plaque to Winston Churchill as he was stationed here, as was Bruce Bairnsfather [the cartoonist].
To the W of the Ypres-Ploegstraat road are a series of craters, including Spanbroekmolen [the "Peace Pool", not far from Wijschate], again as Canardly said, well worth a visit.
Also, about half a mile W of Ypres road north of Wijschate is the Crooneart Wood which contains the Bayernwald trench system. Entry to this is via tickets from the tourist office in Kemmel, which allows access through the perimeter fencing. There is a small charge. Watch out for Kemmelberg ["Mount Kemmel"], it's pretty steep. In fact the area of Heuveland, to the SW of Ypres is quite hilly.
About 4 miles to the W of Ypres, is the Brandhoek New Military Cemetery. Here is buried Captain Noel Chavasse VC and Bar, his grave is right next to the cross. There are several German graves in the cemetery as well.
If you continue W another three miles [avoid the main road], you get to Poperinghe. Here there is Talbot House [home of "Toc H"], in the main square there are Cafe a la Fabrique, La Poupee and Cafe de Esperance, all apparently well known to British troops [look out for the plaque about "Ginger"]. On the E side of the main squre is a building [I think it's the old town hall] which has a courtyard. If you go into this you will see the condemned cell and the execution post.
There are no end of memorials, cemeteries and old bunkers that you will come across, especially if you don't mind following a dirt track or two. Generally the cycling is pretty flat, certanly to the east of Ypres and the Messine Ridge. To the west can be quite hilly in places.
One word of warning, if you're heading from Plugstreet towards Wambrechies [on the road to Lille] don't be tempted to try and cross the river at Pont Rouge. The maps show the bridge as viable but when we got there, after a 70 mile loop from Lille, it was derelict. After a very long and tiring day it was all we needed.
