Holman's (HCSB) has a nice pocket version I still use as a carry Bible. I like the translation more and more as I read it, too!
As for the Psalms, you are totally wrong to think any of them can be left out. When I went through a hard time in my life, God told me to read 5 Psalms a day! They changed my theology, brought me closer to God and through that difficult time. I still read a Psalm every day, (going on 12 years!) in addition to 3 of the OT and 3 of the NT. There is a reason that Gideon's, who give away Bibles, put them in hotel rooms, have a pocket version of the NT which includes the Psalms. Because lonely people in hotel rooms can find hope and encouragement in the Psalms.
You might want to get a pocket version of Gideon's. They are free, but they do miss most of the Old Testament.
Oh, thin paper helps make it smaller. I read my Bible at home, and have used thick, heavy study Bibles for many years now. So I do get what you are saying! You might want to search Amazon, Lifeway or Zondervan (ESV) to see what they carry, including weight and size.
If you are wondering about translations, anything modern and readable is best. ESV, NIV, NASB, HCSB, even NET are good. Stay away from KJV. It really is a different language, and not the best translation in terms of manuscript evidence and using not using understandable English!