Than Angels  
   Angels are "ministering spirits, sent out to render service for the sake of those who will inherit salvation.." Heb 1:14  
God did not say to the angels " Thou art my Son, Today I have begotten Thee" Heb 1:5;(cf Ps 2:7; 2Sam 7:14; 1Chr 17:13), He said this to Christ. Angels are created beings, Christ is begotten! The angels of God worship Christ Heb 1:6; Duet 32:43. It is Christ who sits at the right hand of God (Heb 1:3) for it was to His Son that He said "Sit at My right hand, until I make your enemies a footstool for your feet," Heb 1:10; Ps 110:1, not an angel. It is Christ whose kingdom is everlasting. Heb 1:8; Ps 45:6-7. 

   Jesus is "better" than Moses because Moses was mortal - he was part of God's house that He served in, whereas Jesus is in charge of God's house - the body of Christ.Heb 3:2-6 

 As our High Priest    
   Christ is "better"  than the Aaronic priests because he was tempted in every way as we are but he didn't sin - therefore, when in obedience to the Father Jesus died on the cross as a sacrifice for sin, it wasn't for himself.  The Aaronic priests had to offer a sacrifice for their sins but Jesus offered his for us alone - He was sinless. Heb 4:14-16; 5:1-3.   Through the new covenant He can make things perfect, whereas the Law of Moses couldn't make anything perfect due to the fact that all men are imperfect.  Christ also lives forever, being able to plead our case in heaven itself, where earthly priests die and the Aaronic High Priest had to go into the Most Holy Place only once a year. Heb 7:16-28.  As the Holy Place and the actions of the Priests are only a copy and a shadow of the heavenly ones (cf Heb 9:23), the originals have to be better, because they are not man-made, and are not part of the created world. Heb 8:5; 9:1-11. 
It is better for everyone to be able to enter the Most Holy Place instead of just the High Priest, as this was, and still is, where we meet God. Heb 10:19-20 

The New Covenant   is "better"  because it is based on better things such as Jesus Christ, through whom we get to know God personally, knowing his laws without having to learn them from books (please make sure you continue to read your bible, because we still need to learn about Him, and grow in Him and it is through His word that He speaks to us), and having your sins forgiven and erased. It is by these better promises of this covenant that God Himself undertook to make good the deficiency and to ensure the realization of His purpose - which is that a company of people should be separated from the world, and brought into fellowship with Himself to be His people, to delight in His company and to do His will Heb 8:6,10-12. 

The blood Christ  shed for us is superior than that of the animal sacrifices because of its power to cleanse us from sin, from the inside out - our hearts and souls. Heb 9:12-14 This sacrifice is "better" because He only had to die once and it is effective forever Heb 10:12.  The blood of animals can't take away sins (it only covers it), but Christ's blood can, and through Christ's sacrifice "He has made perfect forever those who are purified from sin" Heb 10:14; 9:25-28; 10:4; 10:9-14. Christ's blood is superior to that of Abel's too. It has the power to forgive sins, heal, and to protect (plus a lot more) and through it being spilt, we are able to be united with Christ as joint heirs of the kingdom of God Heb 12:24.  This is one incredible possession we have that no one can ever take from us,  our salvation, and it is the best possession we can have. Heb10:34. 

Even God's punishment is better than our father's, because it shows us that God cares enough about us to punish us for our own good, so that we can share his holiness, but our own fathers mete out punishment as they see fit. Heb 12:10.  

In closing , we have a better relationship with God, our Father, through Christ, than the Hebrews do through their High Priest. Jesus is always there for us Heb 13:5-16 - all we have to do is pray and we can speak to God through Christ, as Jesus is the way to the Father! John 14:6